# Numerical Investigation on NOx Emission of a Hydrogen-Fuelled Dual-Cylinder Free-Piston Engine
## 2 Simulation Work
### Hydrogen Free-Piston Engine Modelling and Validation
#### 2.3.3 Validation of Heat Release Model  
Heat release rate is a function of viscosity, density, calorific value, latent heat of fuels, burning velocity of fuels and combustion temperature. The heat release rate consists of premixed combustion, controlled combustion and late combustion. The heat release rate is derived from the first law of thermodynamics, assuming it to be an open system. As the heat release rate and cumulative heat release describe the details of the combustion process, it is necessary to calculate these two figures for the two-stroke free-piston engine of Ugo. According to the parameters and data profile provided, the calculation procedure is as follows. The starting point for calculating the heat release rate and cumulative heat release from the indicator diagram is the first law of thermodynamics. When calculating the heat release rate and cumulative heat release, the zero-dimensional combustion model is adopted, and the mixture is considered to be evenly distributed in the cylinder. Part of the heat released by fuel combustion is used to improve the internal energy of the gas in the cylinder and to do external work. Part of the heat is transmitted to the wall of the combustion chamber to form heat dissipation loss. The formula for the heat balance at any moment in the combustion process is:  
\[Q_{f}=Q+Q_{w}=\Delta U+W+Q_{w} \tag{1}\]  
where \(Q_{f}\) is heat released by fuel before this instant; \(Q\) is heat absorbed by in-cylinder gas before this instant; \(Q_{w}\) is heat transferred to the cylinder wall before this instant; \(\Delta U=U-U_{a}\), where \(U_{a}\) is the internal energy of the in-cylinder gas at the beginning of calculation; \(W\) is the work done by the gas from the beginning of calculation to this instant. Within the step length of \(\Delta\varphi\) crank angle: \(\Delta Q_{f}=\Delta U+\Delta W+\Delta Q_{w}\), in order to calculate the heat release \(\Delta Q_{f}\), it could be calculated from the practise test \(p-\Phi\) indicator diagram and obtain internal energy change \(\Delta U\), the work done \(\Delta W\) and heat dissipation \(\Delta Q_{w}\). The goal in defining heat capacity is to relate changes in the internal energy to measured changes in the variables that characterise the states of the system. For a system consisting of a single pure substance, the only kind of work it can do is atmospheric work, and so the first law reduces to:  
\[dU=d^{\prime}Q-PdV \tag{2}\]  
Figure 9: Indicated power comparison between the WAVE model and experimental data. Suppose now that \(U\) is regarded as a function \(U\) (\(T\), \(V\)) of the independent pair of variables \(T\) and \(V\). The differential quantity \(dU\) can always be expanded in terms of its partial derivatives according to:  
\[dU=\left(\frac{\partial U}{\partial T}\right)_{V}dT+\left(\frac{\partial U}{ \partial T}\right)_{T}dV \tag{3}\]  
where the subscripts denote the quantity being held constant when calculating derivatives. Substituting this equation into \(dU=d^{\prime}Q-PdV\) then yields the general expression:  
\[d^{\prime}Q=\left(\frac{\partial U}{\partial T}\right)_{V}dT+[P+\left(\frac{ \partial U}{\partial T}\right)_{T}]dV \tag{4}\]  
for the path-dependent heat. The path can now be specified in terms of the independent variables \(T\) and \(V\). For a temperature change at constant volume, \(dV=0\), by definition of heat capacity,  
\[d^{\prime}Q_{V}=C_{V}dT \tag{5}\]  
The above equation then immediately gives  
\[C_{V}=\left(\frac{\partial U}{\partial T}\right)_{V} \tag{6}\]  
for the heat capacity at constant volume, showing that the change in internal energy at constant volume is entirely due to the heat absorbed.