## 3 Numerical Analysis  
### Physicochemical Properties of Biodiesel  
Biodiesel has promising lubricating properties and cetane number compared to low sulfur diesel fuels [19]. The lower heating value of biodiesel is about 37.4 MJ/kg, which is 9% lower than regular fossil diesel. Biodiesel has higher cetane values, resulting in better ignition. The higher cetane number a fuel possesses, the more easily the fuel can combust in a compression ignition engine. Faster premixed combustion can effectively shorten the ignition delay period, reducing the peak of pressure rise rate. Therefore, higher-cetane fuel is usually considered to cause an engine to run more smoothly and quietly. The basic physical and chemical properties of biodiesel and standard diesel are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Properties**} & **Density** & **Heat of Vaporization** & **Lower Heating** & **Kinematic** & **Surface** & \\  & **(20 °C/(g/cm3)** & **(kJ/kg)** & **Value (MJ/kg)** & **(mm2/s)** & **(mN/m)** & **Cetane** \\ \hline \hline   Diesel & 0.831 & 250 & 42.6 & 2.6 & 31.1 & 55 \\ Biodiesel & 0.884 & 320.2 & 37.4 & 5.2 & 30.4 & 59 \\ \hline \hline  & & & & & & \\ \end{tabular}
\end{table} 
Table 3: Physical and chemical properties of alternative fuels. Figure 7: Comparison of simulated and measured acceleration spectra. In this study, pure biodiesel and pure fossil diesel were prepared and used as the basis for investigating the influences of alternative fuels. The biodiesel was produced from waste cooking oil based on ASTM D6751-09 standard [20]. The in-cylinder combustion pressures of two kinds of fuels at four different operating conditions were acquired from a single-cylinder diesel engine. The measured pressure curves and calculated pressure rise rates are presented in Figure 8.  
Compared with fossil diesel, biodiesel always shows a pre-ignition phenomenon in different operating conditions, as seen in Figure 8. A higher cetane number of biodiesel means faster in-cylinder burning velocity. The faster the premixed combustion, the shorter the ignition delay period and the lower the peak of pressure rise rate. This also explains why biodiesel possesses a lower magnitude in the peak of pressure rise rate than the others. In the operating condition of 1000 rpm engine speed and 40 Nm torque output, both fuels exhibit substantial delays in the peak of combustion pressure. With the engine load going up, the increased fuel delivery per cycle will significantly enhance the heat release of the premixed combustion phase, and enlarge the in-cylinder pressure. In the meantime, the increased proportion of premixed combustion can cause phase delay in the whole combustion process, resulting in the appearance of a significantly delayed pressure peak. As mentioned above, the periodically changing side-thrust force can be decomposed from the longitudinally acting combustion force and inertial force of the moving piston assembly. Therefore, the lateral driving force of the piston that is applied to the cylinder liner is bound to be affected by variations in the combustion force, determined by the combustion characteristics of different fuels. The influence of differences in fuel types on the piston side-thrust force mainly manifests clear differences in its peak value and associated phase. Under low speed conditions, the peak value of the side-thrust force after combustion top dead center (TDC) burning with biodiesel is lower than that with diesel, as seen in Figure 9a,b, which might be attributed to the lower pressure peaks. In the high-speed condition of 1800 rpm and 10 Nm, the peak value of biodiesel is slightly larger than that of diesel. Under high-speed conditions, the peak values of side force burning with biodiesel are close to those of diesel. This is because biodiesel has lower energy content than diesel fuel.