# Temperature Dependence of Density and Viscosity of Biobutanol-Gasoline Blends
## 2 Materials and Methods
### Mathematical Models for the Density, Dynamic, and Kinematic Viscosity  
In this paper, four mathematical models were used for the temperature dependence of the volumetric mass density and both viscosities. These models are used for nonlinear (frequently exponential) dependences, which is also observed in the temperature dependence of the volumetric mass density and the viscosity of the fuels.  
Some of models below were used also in papers [26; 27; 28; 29; 30]. The details and formulas of the used mathematical models are shown below.  
Exponential model:  
\[\eta=\eta_{0}\cdot e^{h\cdot T}\text{ for dynamic viscosity} \tag{2}\]  
\[\eta=\eta_{0}\cdot e^{h\cdot T}\text{ for dynamic viscosity} \tag{3}\]  
\[\rho=\rho_{0}\cdot e^{h\cdot T}\text{ for volumetric mass density} \tag{4}\]  
[PERSON] model:  
\[\eta=\eta_{0}\cdot e^{\frac{h}{h\cdot T}}\text{ for dynamic viscosity} \tag{5}\]  
\[\
u=\
u_{0}\cdot e^{\frac{h}{h\cdot T}}\text{ for kinematic viscosity} \tag{6}\]  
[PERSON] model:  
\[\eta=\eta_{0}\cdot e^{\frac{h}{h}}\text{ for dynamic viscosity} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Property** & **Unit** & **Gasoline** & **Bioethanol** & **Biobutanol** \\ \hline Chemical formula & & C\({}_{4}\) to C\({}_{12}\) & C\({}_{2}\)H\({}_{5}\)OH & C\({}_{4}\)H\({}_{9}\)OH \\ Molecular mass & kg\(\cdot\)mol\({}^{-1}\) & 109 & 46 & 74 \\ Volumetric mass density (15 \({}^{\circ}\)C) & kg\(\cdot\)m\({}^{-3}\) & 720–775 & 789 & 814 \\ Calorific value & MJ\(\cdot\)kg\({}^{-1}\) & 43.3 & 29 & 33 \\ Heat of vaporization & kJ\(\cdot\)kg\({}^{-1}\) & 349 & 918 & 583 \\ Octane number ROM & - & 91–99 & 107 & 96 \\ Boiling point & \({}^{\circ}\)C & 27–225 & 78 & 104 \\ Flash point & \({}^{\circ}\)C & \(-\)45 & 13 & 35 \\ Auto ignition Temp. & \({}^{\circ}\)C & 258 & 420 & 415 \\ Stoichiometric AFR & - & 14.7 & 9.01 & 11.12 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of selected properties of fuels [22; 23].  
\[\
u=\
u_{0}\cdot e^{\frac{b}{T}}\text{ for kinematic viscosity} \tag{8}\]  
Vogel model:  
\[\eta=\eta_{0}\cdot e^{\frac{b}{T+T}}\text{ for dynamic viscosity} \tag{9}\]  
\[\
u=\
u_{0}\cdot e^{\frac{b}{T+T}}\text{ for kinematic viscosity} \tag{10}\]  
where \(\eta\) is the dynamic viscosity, \(\eta_{0}\) is the reference value of the dynamic viscosity, \(\
u\) is the kinematic viscosity, \(\
u_{0}\) is the reference value of the kinematic viscosity, \(\rho\) is the volumetric mass density, \(\rho_{0}\) is the reference value of the volumetric mass density, \(T\) is the absolute temperature, \(R\) is the Universal gas constant, \(E_{a}\) is the Arrhenius activation energy, and \(b\) and \(c\) are coefficients.  
Furthermore, mathematical polynomial multivariate models were created. These models describe the behaviour of the volumetric mass density or kinematic viscosity of the biobutanol-gasoline blends depending on their temperature and concentration. The models were created for both the RON 95 and RON 98 gasolines. The equations of the multivariate models are given below based on polynomial functions.  
The volumetric mass density:  
\[\rho(T,C)=x_{0}+x_{1}\cdot T+x_{2}\cdot C+x_{3}\cdot T^{2}+x_{4}\cdot T\cdot C \tag{11}\]  
The kinematic viscosity:  
\[\
u(T,C)=y_{0}+y_{1}\cdot T+y_{2}\cdot C+y_{3}\cdot T^{2}+y_{4}\cdot T\cdot C. \tag{12}\]  
In Equations (11) and (12), \(\rho\) is the volumetric mass density, \(\
u\) is the kinematic viscosity, \(T\) is the absolute temperature, \(C\) is the concentration, and \(x_{i}\), \(y_{i}\) are coefficients.