**1. Introduction**

Diesel engine combustion is an unsteady and discontinuous process, in which the fuel chemical energy is transferred to the work medium internal energy. Due to its complicated energy conversion process, modeling the diesel engine combustion is quite challenging to both engine designers and users [1]. When the engine is applied in an integrated system such as the diesel engine control system, after-treatment system, ship propulsion system, etc., the combustion details are not the main concern, but the main performance parameters of the engine, such as engine work, peak pressure, peak temperature, are strongly affected by the engine combustion process. A suitable approach on fitting diesel engine combustion process with a few parameters is useful for combustion modeling, where the mathematics analysis is essentially required to solve the rooting finding problems in particular for the experimental investigation, because plenty of experimental data needs to be handled and used in the multi-dimensional system of equations [2,3].

Diesel engine combustion process is varied in comparison with the other combustion machines, resulting in the specific theoretical investigation method. Recently there has been lots of diesel engine combustion research using both theoretical and experimental approaches. With the rapid development of three-dimension numerical calculation technology, the diesel engine combustion can be simulated in quite detailed ways to obtain the detailed combustion information [4,5]. A three-dimension simulation of diesel engine combustion requires the turbulence flow model, spray model and combustion model, among which the turbulence flow model plays an important role. Normally there are three ways to model the turbulence flow of the engine: Large Eddy Simulation (LES), Direct Numerical Simulation (DNS) and Reynolds-Averaged Navier-Stokes (RANS) [6]. Kahila et al. developed the LES together with a finite-rate chemistry model to investigate the engine dual-fuel ignition process [7]. Knudsen et al. proposed the compressible Eulerian numerical model to describe how liquid fuel injector nozzle geometry and operation strategies influence gas phase fuel distribution [8]. An et al. used the renormalization group K-epsilon model to describe the turbulent flow field and predict the soot particles evolution [9]. When modeling the turbulent flow, it is important to calculate the average chemical reaction rate and the turbulent combustion rate of fuel [10,11]. Yu et al. used the characteristic time combustion model coupled with the chemical kinetic mechanism to study the mixture formation's influence on the smoke [12]. Cheng et al. studied the methyl esters of soybean and coconuts formed by spray combustion and related emissions of the diesel engine and further to the effect of EGR on these biodiesel fuel combustion [13]. The diesel engine spray models are mainly divided into homogeneous gaseous jet models and gas-liquid two phase models, and the latter use the gas and liquid two phases in modeling engine spray process [14–16].

Besides the three-dimension approaches in modeling the engine combustion process, the multi-zone models are also efficient tools. Compared with the commonly used single-zone models, the multi-zone models take into account the instantaneous details of combustion process, such as the formation and development of fuel spray, the relative motion of fuel droplets and air. A popularly used multi-zone combustion model is Hiroyasu's fuel droplet evaporation combustion model [17], which includes the fuel injection sub-model, combustion thermodynamic calculation sub-model and emission generation sub-model. Zhang et al. used the in-cylinder steam injection method to establish a two-zone combustion model to research the waste heat recovery and NOx emission control [18]. Fang et al. built a two-dimension combustion model to study the effect of electric fields on the combustion characteristics of a lean-burning methane air mixture [19]. Xiang et al. proposed a two-zone combustion model in nature gas engine application and used this model for engine knocking predictions [20].

The diesel engine combustion measurement techniques include visualization, PIV (particle image velocimetry), LIF (laser induced fluorescence) and PDPA (phase Doppler particle anemometry). The PIV velocity measurement is a non-contact, transient and full-flow velocity measurement method, which has the advantages of not interfering with the quantitative information of the measured field and fast dynamic response [21,22]. LIF method produces relatively strong light signals with high spatial resolution and the fluorescence intensity is proportional to the incident light intensity so that it can be used to measure the concentration of substances [23]. The PDPA method depends on the frequency difference between the scattered light and the irradiated light of the moving particles, whose size is determined by analyzing the phase shift of the scattered light reflected or refracted by the spherical particles passing through the laser measuring body [24]. The diesel engine combustion experimental research supplemented with the theoretical investigation improve the engine combustion performance and reduces emissions.

Although the detailed engine combustion models are developed rapidly with the progress of simulation tool and experimental facilities, when the engine is used as a component of systems such as the control system, after treatment system and propulsion system, the engine combustion process has to meet the requirement of the system real-time characteristics. Mean value model is a time-domination model with cyclic time scale instead of the crank angle time scale, having the characteristics of fast calculation to ge<sup>t</sup> the engine main performance parameters, and it is commonly used in the control system and real-time dynamic simulation. Yin et al. introduced a turbocharged diesel engine model for hardware-in-the-loop simulation, which has the capability of observing engine state parameters and capturing engine transient response [25]. Sui et al. introduced a two finite stages Seiliger process model and simulates the engine combustion process by several parameters, which were applied in the ship propulsion system simulation [26]. Baldi et al. combined the zero-dimension and mean value

models to calculate the engine performance parameters including the in-cylinder ones in relatively short simulation time [27]. Theotokatos et al. developed an extended engine mean value model to predict the thermodynamic parameters of two-stroke marine engine at different injection times [28]. Meanwhile, plenty of "black-box" methods such as multi-level hierarchical, neural networks, fuzzy logic, and wavelet networks are also applied in the modeling engine combustion process [29–31].

During the modeling engine combustion process, the experimental methods are frequently used together with theoretical analysis to parameterize the combustion process, in which the mathematics analysis is necessary to find the roots in the systems of equations deriving from the balance in the engine combustion process. Heat release rate calculation based on the measured in-cylinder pressure, energy conservation equation and empirical heat transfer formula is a fundamental method to obtain the engine combustion phenomena. On the other hand, when the heat release is known, the statistical analysis of a large amount of actual heat release can be used to build the heat release models based on the empirical formula or curve fitting [32]. The general semi-empirical formula for calculating heat release rate is Vibe formulas and the isosceles triangle combustion model [33,34]. In addition, if one only needs to know the engine in-cylinder main operating parameters under various conditions instead of the engine combustion details, the Seiliger process is the appropriate method for engine combustion modeling [26,35,36].

Nevertheless, it is critical to the mean value simulation model research that the derivation of Seiliger parameters such as the combustion parameters to give a global description of the combustion process obtains lots of experimental data or a deep theoretical investigation. Most of the researchers merely focused on obtaining the Seiliger parameters according to the engine experiments and ignoring the mathematics applications during the fitting process, resulting in the low accuracy. On the other hand, the Seiliger parameters of only one engine operating point, normally the nominal operating point, were used in these literatures, and the others were obtained based on the mathematical interpolation methods.

In the case of getting Seiliger parameters based on the combustion fitting process, the multiple variables non-linear system of equations has to be set up with equivalence criteria. When solving non-linear equations in engineering problems, the Bisection method, Secant method, Newton-Raphson method, etc. are usually used [37,38]. Sciniaka et al. introduced several iterative algorithms for solving equations and proposed an adaptive method based on the Newton-Raphson rooting method, which can better control the algorithm dynamics [39]. Furthermore, when more variables are chosen, the system of equivalence criteria becomes a multi-variable function and the Newton-Raphson method is a suitable way to find the roots of the system of equations. The Newton-Raphson method has a prominent advantage in that it has a square convergence around a single root of the equation *f*(x) = 0. It can also be used to find the multiple roots and the complex roots of the equation [40,41]. Meanwhile, the method converges linearly but can be super linear convergence by some conversions.

In this paper, an experimental investigation into combustion fitting is carried out based on the in-cylinder pressure signals measurement. With the processed measurement data, the Seiliger process model is introduced, and after determining the Seiliger parameters and equivalence criteria, the diesel engine measurement is fitted with Seilgier process model for engine combustion analysis. The fitting results of engine running at nominal point are shown in both single cylinder and four-cylinder average signals, in which the discrepancy of cylinder working condition can be obtained. Last but not least, the results of engine running at constant speed are discussed for further research on combustion fitting models on the systematic simulation of marine diesel engine applications.
