# Low Temperature Combustion Modeling and Predictive Control of Marine Engines
## 2 Materials and Methods
### Nonlinear RCCI Control-Oriented Model
#### 2.1.2 Burn Duration Model  
According to [34], the primary mechanism in RCCI combustion involves the formation and propagation of ignition pockets within the charge. This enables the utilization of spontaneous ignition front speed (\(S_{ig}\)) for estimating the burn duration (\(\theta_{\text{BD}}\)). [PERSON] et al. [13] formulated a correlation between burn duration and \(S_{ig}\) as follows  
\[\theta_{\text{BD}}=\alpha_{1}(S_{ig})^{\alpha_{2}}, \tag{4}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are tuning coefficients. Here, \(S_{ig}\) is defined as  
\[S_{ig}=\frac{1}{|\
abla\tau|}. \tag{5}\]  
According to Equation (5), the subsonic spontaneous ignition front speed is dependent on the gradient of the ignition delay [35]. The ignition delay gradient of the mixture does not solely rely on the initial state; it is influenced by the extent of compression heating during combustion as well. This heating, in turn, hinges on the initial distribution of the mixture conditions [35]. As Equation (5) indicates, the \(S_{ig}\) is a function of the gradient of the ignition delay. It should be noted that the gradient of \(\tau\) can be considered in terms of one or several variables highlighted in Equation (1). The study of [36] included all of the variables in the gradient of ignition delay. In [13] ignition front speed is estimated based on the changes of the equivalence ratio. According to their study, the gradient of ignition delay is attributed to the stratification of equivalence ratio. Their formulation of \(S_{ig}\) in Equation (5) for an RCCI engine is as follows:  
\[S_{ig}=\frac{1}{|\frac{\partial\tau}{\partial\phi_{\text{SOC}}}||\
abla\phi_{ \text{SOC}}|}, \tag{6}\]  
where the ignition delay (\(\tau\)) is indeed defined as the denominator of the second integral in Equation (1) and \(\
abla\phi_{\text{SOC}}\) is the gradient of equivalence ratio at the start of combustion. In this work, variations in the injected fuel mass are considered as the source of the gradient in \(\phi\). [PERSON] et al. [13] employed the following assumptions to establish the gradient of the equivalence ratio within the charge:  
* The gradient of equivalence ratio remains uniform with respect to the radial distance from the cylinder's centerline, particularly from the injection point. The highest diesel equivalence ratio is typically observed in close proximity to the cylinder liner and diminishes steadily as the radial distance towards the cylinder centerline increases [34]. This assumption leads to a simplified one-dimensional distribution of equivalence ratio across the mixture. * Estimating the maximum equivalence ratio involves considering the mixing degree of diesel fuel, which is contingent on the ignition delay. Given that the start of combustion has been previously estimated using Equation (1), the apparent ignition delay (\(\theta_{\text{ID}}\)) in crank angle degrees (CAD) is computed as shown in Equation (7). \[\theta_{\text{ID}}=\theta_{\text{SOC}}-\theta_{\text{SOI}}.\] (7)  
Given the apparent ignition delay, estimating the maximum equivalence ratio value for HRF (\(\phi_{\text{HRF,max}}\)) that aligns with the maximum value of equivalence ratio within the mixture is accomplished by:  
\[\phi_{\text{HRF,max}}=\frac{\beta_{1}}{\theta_{\text{ID}}^{\beta_{2}}}\phi_{ \text{HRF}}^{\beta_{3}}. \tag{8}\]  
Here, \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) are tuning coefficients. The derivative of \(\tau\) with respect to \(\phi\) arises from the linear spatial distribution of the diesel fuel.