# Low Temperature Combustion Modeling and Predictive Control of Marine Engines
## 2 Materials and Methods
### Nonlinear RCCI Control-Oriented Model
#### 2.1.1 Modified Knock Integrated Kinetics Model  
The modified knock integral model (MKIM) is a modified version of the Livengood and Wu knock integral model (KIM) [32], originally developed to predict auto-ignition in  
Figure 1: Flowchart of the developed RCCI control-oriented combustion model for predicting main combustion indicators.  
spark ignition (SI) engines. [PERSON] [33] modified the correlation to predict auto-ignition for HCCI combustion. The model was eventually modified towards HCCI combustion by [PERSON] et al. [13], which is presented in Equation (1). The original KIM was adjusted towards RCCI based on polytropic compression, fuel concentration, premixed ratio and diesel injection timing. The first integral in Equation (1) describes the compression of LRF entering the combustion chamber and mixing with air from \(\theta_{\text{IVC}}\) to \(\theta_{\text{SOI}}\) and the second integral describes the compression of the mixture of HRF and LRF from \(\theta_{\text{SOI}}\) to \(\theta_{\text{SOC}}\) leading to the auto-ignition. The start of combustion occurs when the integral approaches unity.  
\[\int_{\theta_{\text{IVC}}}^{\theta_{\text{Soc}}}\frac{1}{A_{1}N \phi_{\text{LRF}}^{B_{1}}\text{exp}\left(\frac{\text{C}_{1}(p_{\text{IVC}} \phi_{c}^{\pi_{c}})^{D_{2}}}{T_{\text{IVC}}\eta_{c}^{\pi_{c}-1}}\right)}d\theta \tag{1}\] \[+\int_{\theta_{\text{SOI}}}^{\theta_{\text{SOC}}}\frac{1}{A_{2}N (\phi_{\text{LRF}}^{B_{2\text{MF}}}+\phi_{\text{LRF}}^{B_{2\text{LRF}}}) \text{exp}\left(\frac{\text{C}_{2}\
abla_{\text{mix}}+b}{T_{\text{IVC}}\eta_ {c}^{\pi_{c}})^{D_{2}}}}{T_{\text{IVC}}\eta_{c}^{\pi_{c}-1}}\right)d\theta=1.\]  
In Equation (1), \(A_{1}\), \(B_{1}\), \(C_{1}\), \(D_{1}\), \(A_{2}\), \(B_{2,\text{LRF}}\), \(B_{2,\text{HRF}}\), \(C_{2}\), \(b\) and \(D_{2}\) are tuning coefficients, such that the condition of the equation holds. The denominator of the second integral (\(\theta_{\text{SOI}}\to\theta_{\text{SOC}}\)) is the ignition delay (\(\tau\) [\(s\)]) corresponding to the mixture of air with both LRF and HRF. Additionally, \(N\) denotes the engine speed, \(\phi\) is the equivalence ratio of HRF and LRF, \(p_{\text{IVC}}\) and \(T_{\text{IVC}}\) are pressure and temperature at IVC, \(n_{c}\) is the polytropic coefficient, \(\text{CN}_{\text{mix}}\) describes the reactivity of the fuel blend and \(v_{c}\) is the ratio of volume at IVC (\(V_{\text{IVC}}\)) to the volume at any crank angle (\(V(\theta)\)). \(\text{CN}_{\text{mix}}\) and \(v_{c}\) are defined in Equations (2) and (3), respectively.  
\[\text{CN}_{\text{mix}} =\frac{\text{FAR}_{\text{st,LRF}}\phi_{\text{HRF}}\text{CN}_{ \text{HRF}}+\text{FAR}_{\text{st,LRF}}\phi_{\text{LRF}}\text{CN}_{\text{LRF}}} {\text{FAR}_{\text{st,HRF}}\phi_{\text{HRF}}+\text{FAR}_{\text{st,LRF}}\phi_ {\text{LRF}}}, \tag{2}\] \[v_{c} =\frac{V_{\text{IVC}}}{V(\theta)}. \tag{3}\]  
In Equation (2), \(\text{FAR}_{\text{st}}\) is the stoichiometric fuel-air-ratio and CN is cetane numbers of the fuels. In this work value of \(n_{c}\) was kept constant, which is same as for the UVATZ model. The selection is based on the criterion of matching net heat released. The values of \(\text{CN}_{\text{HRF}}\) and \(\text{CN}_{\text{LRF}}\) are based on surrogate fuels in the UVATZ model. More details about the model and its formulation can be found in [13].