^{1}

^{*}

^{1}

^{2}

^{2}

^{3}

Diesel engines have proven over the years important in terms of efficiency and fuel consumption to power generation ratio. Many research works show the potential of biodiesel as a substitute for conventional gasoil. Mainly, previous and recent researches have focused on experimental investigation of diesel engine performance fuelled by biodiesel. Researches on the mathematical description of diesel engine process running on biodiesel are scarce, and mostly about chemical and thermodynamic description of the combustion process of biodiesel rather than performance studies. This work describes a numerical investigation on the performance analysis of a diesel engine fuelled by palm oil biodiesel. The numerical investigation was made using a semi empirical 0D model based on Wiebe’s and Watson’s model which was implemented via the open access numerical calculation software Scilab. The model was validated first by comparing with experimental pressure and performance data of a one cylinder engine at rated speed and secondly by comparing with a six cylinders engine performance data at various crankshaft rotational speeds. Simulations were then made to analyze the engine performance when running on biodiesel. The calculations were made at constant combustion duration and constant coefficient of excess air. Results showed that the model matches the overall experimental data, such as the power output and peak cylinder pressure. The ignition delay was somehow underestimated by the model for the first experiment, which caused a slight gap on in cylinder pressure curve, whereas it predicted the average ignition delay fairly well for the second set of validation. The simulations of engine performance when running on biodiesel confirmed results obtained in previous experimental researches on biodiesel. The model will be further investigated for engine control when shifting to biodiesel fuel.

Reaching lower toxicity of exhaust gases, whilst reducing fuel consumption is one of the modern trends in automobile engineering research. The characteristics of fuel play an important role in the performances of the engine. Gasoil has been for years the main fuel for diesel engine but, due to the reducing availability of fossil energy resources and the stricter rules on engine emission [

Simulation and mathematical modeling of diesel engine are scientific topics carried out by several research works. Nowadays, there mainly exist three types of approaches for diesel engine simulation: 0 dimensional thermodynamic, quasi dimensional and multi dimensional (Computational Fluid Dynamic). A well documented discussion on these models can be found in [

Many commercial computer codes based on computational fluid dynamic (CFD) [

The model we implemented in our study is based on the engine performance Wiebe model [

One of the limitations of the Wiebe model is that it doesn’t predict well the combustion during the premix phase of combustion. That limitation is overcome by Watson model of heat release in diesel engine [4,10]. Both Wiebe and Watson model don’t take into account the ignition delay of the injected fuel in the cylinder, thus obliging us to find a suitable relationship to predict ignition delay. Pischinger et al. [

The present work, describes a semi empirical model used to predict how performances parameters such as thermal efficiency, specific fuel consumption, indicative pressure and indicative pressure will vary depending type of fuel used. The model is developed Fuel for diesel engine control purpose when running on biodiesel. Spray behavior of biodiesel is not covered in this work but some useful computational studies about the topic can be found in [14,15], and it will be inserted in further development of the model.

The model used in the present study is a semi-empirical model, based on the work of I. I. Wiebe [

The initial Wiebe model was not taking into account the ignition delay period when calculating the combustion process. Wiebe model computes the heat release starting from the injection start angle, whereas it has been shown that ignition occurs after a certain amount of time after injection (ignition delay), due to complex chemistry processes.

In our model we used the model proposed by Hardenberg and Haze [

,

where U_{P} is the mean piston speed in m/s; R is the gas constant in J/kmol-K; ε is the compression ratio of the engine; CN is the cetane number of the fuel P_{im} and T_{im} are the pressure in bar and temperature in Kelvin at the intake manifold; n_{c} is the polytropic exponent for compression.

The next step of the model is the fuel engine cycle model; here we determine the pressure inside the cylinder at any angle of rotation of the crankshaft taking into account the start of injection angle the duration of combustion and other parameters. As a result one will be able to determine the different parameters characterizing the efficiency of work of the engine, such as the specific consumption, the effective power and efficiency. The model computes the full process taking place inside the cylinder, with the calculations being made for each stroke of engine cycle.

The pressure of the working medium at the end of the admission stroke is given in Pascal by

,

where η_{υ} is the admission coefficient, P_{k} is the pressure at inlet valves in MPa, T_{k} the temperature at inlet valves in K, ΔT is the temperature gradient due to the heating of engine elements in MPa, P_{r} is the residual gases pressure in MPa.

The temperature of the working medium at the end of the admission stroke is given in Kelvin by

,

where T_{k}_{ }_{ }is the temperature of residual gases in K, ς is the coefficient of residual gases.

The theoretically necessary (stoichiometric) quantity of air for the combustion of 1 kg of fuel is given as, its value is dependent of the chemical composition of the fuel used and is determined by

,

where, C, H and O are respectively, the ratio of carbon, hydrogen and oxygen in the fuel chemical composition.

The specific volume of the working medium at the end of the admission stroke is given in m^{3}/kg by

,

where μ_{air} is the molecular mass of air.

The parameters of the working medium during the compression stroke are computed using the polytropic process equation.

The pressure at a given time is given in MPa by

where ν is the current value of specific volume defined as

,

σ is the kinematic function of the motion of the piston

,

where λ is the ratio of the lengths of the crankshaft and the connecting rod, φ is the current angle of rotation of the crankshaft.

The specific work of compression is then determined in MJ/Kg by

The admission, compression and ignition delay phase being computed, the next is step of the model is the heat release calculation. Our heat release model computes two phases of the combustion process, the premixed and diffusion phase.

The current fraction of fuel burnt—where φ is the current crankshaft position angle—is computed using a double Wiebe function [

,

with x_{p} and x_{d} representing the fraction of fuel burnt in each phase of the combustion process, β representing the fraction of fuel injected during the premixed phase.

For each phase of the combustion phase (equation 10) we can write

,

where ai and m_{i} are experimental shaping coefficient of the Wiebe function; φ_{comb} is the start of ignition angle; is the combustion duration for each phase.

The normalized combustion rate (1/deg) is computed by derivation of x_{i} about φ.

The combustion effectiveness which accounts for heat loss (heat loss due to heat transfer to the walls, hydraulic losses due to the flow of gases) ratio is first defined by

where δ is a heat release factor which takes into account the ration of unburned fuel; ψ—is a ratio of used heat.

The total specific heat of combustion used (instantaneous heat release) is given in MJ/Kg by

,

where H_{u} is the net calorific value of the used fuel in MJ/Kg and α is the coefficient of excess air and γ is the ratio of specific heat during the combustion process, according to [

Pressure calculation at any moment of the power stroke is computed using the first law of thermodynamic and can described for the evolution of the pressure/volume indicator diagram from a point 1 to 2 by

,

where: q_{1-2} specific heat used to increase the internal energy from point 1 to 2 in the diagram; C_{v}_{1-2} is the average specific heat the working medium for constant pressure from point 1 to 2.

Assuming each volume step is small enough and using the trapezoidal method to simplify the integral in formula (14) and expressing C_{v}_{1-2} in term of P_{2} using the Mayer’s formula, we determine the value of the pressure at any given time of the power stroke using the simplified equation

,

Δx_{1-2} is the ratio of fuel burnt from point 1 to 2.

Specific work of gases during the combustion stroke is given in MJ/Kg by

.

Further description of the model can be found in [

The validation of the model consisted in comparing experimental results from previous researches with simulation results using the model. For this purpose, experimental data obtained by Sahoo et al. [

The simulation were performed on a 3 Go of RAM Dual core computer with a time step of 0.1 crank angle degrees, a full simulation took about 6 minutes to complete. The parameters of the model were adjusted to closely match the experimental results. The diffusive combustion duration was estimated about 75 and 60 degrees of rotation of the crankshaft for the first and second experiment respectively using least square fitting technique.

Tables 1 and 2 present the engines specifications for each experiment. The default injection timing for the second experiment was determined from the fuel line pressure diagram. The second experiment was performed under varying crankshaft rotational speed—1000, 1250, 1500, 1750 and 2000 rpm.

The calorific value of the diesel fuel was also modified for the second experiment to 42.5 MJ/kg to match experimental input data, remaining properties of the fuel were kept unchanged. To perform comparison we used fuel characteristics that were stated in the work of Sahoo et al. [