# Heat Release Rates of Straight Soybean and Diesel Oil Blends in a Compression Ignition Engine
## 2 Experimental Procedure
### Experimental Heat Release Rate  
The post-processing of pressure measurements is known as the heat release rate analysis, and it is performed during the compression and expansion strokes, while the valves remain closed. The combustion chamber is modeled according to zero-dimensional modeling proposed by [PERSON] and [PERSON] [34] applied to the control volume defined by the engine cylinder. The only mass flow across the boundary is the fuel injected, assuming no leaks through possible gaps in the cylinder. The energy and the mass conservation equations for this control volume are  
\[\frac{d(mu)}{dt}=\frac{dQ}{dt}-p\frac{dV}{dt}+h_{f}\frac{dm_{f}}{dt} \tag{1}\]  
\[\frac{dm}{dt}=\frac{dm_{f}}{dt} \tag{2}\]  
where \(dQ/dt\) represents the heat transfer rate through system boundaries, \(pdV/dt\) is the work rate performed by the system due to system boundary displacement, \(dm/dt\) is the mass flow rate across the system boundary, \(u\) is the specific internal energy of the gas inside the system boundary, \(h_{f}\) is the fuel enthalpy, and \(dm_{f}/dt\) is the mass fuel flow rate. Following the hypothesis that fuel injection is the only mass changing in the cylinder and that the single combustion model assumes a complete burning instantaneously as it enters the combustion chamber, and also that the thermodynamics states of the working fluid are affected by the chemical energy release, the term \(dm_{f}/dt\) can be considered as the mass burning rate of the fuel. The equivalence ratio \(\phi\) at each instant is  
\[\frac{d\phi}{dt}=\frac{1}{m_{a}(FA)_{s}}\frac{dm_{f}}{dt} \tag{3}\]  
where \((FA)_{s}\) corresponds to the stoichiometric fuel/air ratio and \(m_{a}\) to the mass air. Since the properties of the gases in the cylinder are a function of \(T,p\), and \(\phi\), the internal energy \(u\) and the gas constant \(R_{g}\) can be expressed as  
\[\frac{du}{dt}=\frac{\partial u}{\partial T}\frac{dT}{dt}+\frac{\partial u}{ \partial p}\frac{dp}{dt}+\frac{\partial u}{\partial\phi}\frac{d\phi}{dt} \tag{4}\]  
\[\frac{dR_{g}}{dt}=\frac{\partial R_{g}}{\partial T}\frac{dT}{dt}+\frac{ \partial R_{g}}{\partial p}\frac{dp}{dt}+\frac{\partial R_{g}}{\partial\phi }\frac{d\phi}{dt} \tag{5}\]  
The equation of state is also considered in the analysis as an ideal gas mixture. Thus, in a differential form,  
\[\frac{1}{V}\frac{dV}{dt}+\frac{1}{p}\frac{dp}{dt}=\frac{1}{T}\frac{dT}{dt}+ \frac{1}{R_{g}}\frac{dR_{g}}{dt}+\frac{1}{m}\frac{dm}{dt} \tag{6}\]  
The temperature variation as a function of time and the fuel burning rate is obtained from Equations (1)-(6). Thus,  
\[\frac{dT}{dt}=\frac{\left[\frac{1}{V}\frac{dV}{dt}+\left(\frac{1}{p}-\frac{1}{ R_{g}}\frac{\partial R_{g}}{\partial p}\right)\frac{dp}{dt}-\left(\frac{1}{R_{g} m_{a}(FA)_{s}}\frac{\partial R_{g}}{\partial\phi}+\frac{1}{m}\right)\frac{dm _{f}}{dt}\right]}{\frac{1}{T}+\frac{1}{R_{g}}\frac{dR_{g}}{\partial T}} \tag{7}\]  
\[\frac{dm_{f}}{dt}=\frac{\frac{dQ}{dt}-\left(\frac{m}{V}D+p\right)\frac{dV}{dt} -\left[mD\left(\frac{1}{p}-\frac{1}{R_{g}}\frac{\partial R_{g}}{\partial p} \right)+m\frac{\partial u}{\partial p}\right]\frac{dp}{dt}}{u-h_{f}+\frac{m}{ m_{a}(FA)_{s}}\frac{\partial u}{\partial\phi}-D-\frac{mD}{R_{g}m_{a}(FA)_{s}} \frac{\partial R_{g}}{\partial\phi}} \tag{8}\]  
where  
\[D=\frac{\partial u}{\partial T}\left(\frac{1}{T}+\frac{1}{R_{g}}\frac{ \partial R_{g}}{\partial T}\right)^{-1} \tag{9}\]  
The volume variation is calculated from the geometrical relations of the engine. The pressure values and the corresponding first derivative were obtained from experimental data collected from the in-cylinder pressure readings. The thermodynamic properties and the corresponding partial derivatives related to the temperature, pressure, and equivalence ratio are determined by using the routines proposed by [PERSON] and [PERSON] [41].