Simple Finite Heat Release Model (SI Engine) Introduction In the following, a finite burn duration is taken into account, in which combustion occurs at θ soc (Start Of Combustion), and continues until θ eoc (End Of Combustion). Figure 1 is a representation of this cycle. The peak pressure will not be as high as the Otto cycle which has a "delta" function heat release. The finite heat release model assumes that the heat input Q in is delivered to the cylinder over a finite crank angle duration. Q in can be estimated considering the expression: Q in = m c k = m a /α k = (C ρ λ v )/α k where: m c = mass of combustible m a = mass of air C = single cylinder capacity ρ = air density (at intake pressure - 1.2 kg/m 3 at 1 bar) λ v = volumetric efficiency α = air-fuel ratio (gasoline stoichiometric air-fuel ratio 14.65) k = lower heating value (for gasoline 44000 kj/kg) θ soc θ eoc Figure 1: Heat Release Model Derivation of Pressure versus Crank Angle for Finite Heat Release The differential first law for this model for a small crank angle change, dθ, is: Using the following definitions, Q = heat release, W = PdV and du = mc v dt, results in: The ideal gas equation is PV = mrt, so and
The first law now becomes Further reducing the equation: Using R = c p c v and k = c p /c v, to define, the energy equation after rearrangement becomes: or If we know the pressure, P, the volume, V, dv/dθ and the heat released gradient, Q/dθ, we can compute the change in pressure, dp/dθ. Alternatively, we can use experimental data for the pressure, P and the volume, V, to determine the heat release term by solving for Q/dθ. First, the volume, V and dv/dθ have to be defined. Both terms are only dependent on engine geometry: V=V cc +A pist s where V cc is the volume of the combustion chamber, A pist is the area of the piston crown and s expresses the piston motion. So taking the derivative of V with respect to the crank angle, θ, results in: dv/dθ = A pist ds/dθ For heat release term, Q/dθ, the Wiebe function for the burn fraction is used. m Where: f = the fraction of heat added θ = the crank angle θ 0 = θ soc = angle of the start of the heat addition (Start Of Combustion) θ = θ eoc θ soc = the duration of the heat addition (length of burn) a = usually 6 m = usually 3 At the beginning of combustion, f = 0, and at the end the fraction is almost 1. The heat release, Q/dθ, over the crank angle change, θ, is: where Q in is the overall heat input.
Taking the derivative of the heat release function, f, with respect to crank angle, gives the following definition of df/dθ. ma m 1 If θ soc θ θ eoc, df/dθ = 0. So now with Q/dθ and dv/dθ defined, the pressure as a function of the crank angle can be solved: P(θ n ) = P(θ n-1 ) + dp/dθ n (θ n θ n-1 ). Figure 2: Wiebe function Figure 3: Derivative of the Wiebe function
Whole pressure curve Once the heat release phase (combustion phase) is defined, the other phases of the thermodynamic cycle have to be taken into account in order to derive the whole pressure curve. In particular: 1. Intake phase 2. Compression phase 3. Expansion phase 4. Exhaust phase have to be modeled. 1. The intake phase is simply modeled setting the pressure value equal to a constant value. If no turbocharged engines are considered the pressure value is equal to the ambient pressure (1 bar), otherwise the pressure at the turbocharger/intercooler compressed air outlet has to be considered. 2. The compression phase is modeled considering a general polytropic transformation PV kc = P sc V sc kc = Const. (typical exponent kc= 1.35). P sc and V sc are pressure and volume at the beginning of the compression phase, respectively. 3. The expansion phase is modeled considering a general polytropic transformation PV ke = P se V se ke = Const. (typical exponent ke = 1.25). P se and V se are pressure and volume at the beginning of the expansion phase, respectively. 4. The exhaust phase is simply modeled setting the pressure value equal to a constant value. If no turbocharged engines are considered the pressure value is equal to the ambient pressure (1 bar), otherwise the pressure at the turbocharger exhaust gas inlet has to be considered. Figure 4: Pressure curve
Simple Finite Heat Release Model (Diesel Engine) The double Wiebe function is an extension of the model used for spark-ignition engines, in order to describe the premixed and diffusive combustion periods observed in diesel engines. The mass fraction of burnt gases x b can be written as where x p is the mass fraction of fuel burnt in the premixed combustion period, x di is the mass fraction of fuel burnt in the diffusive combustion period, θ p and θ di are, respectively, the duration of premixed and diffusive combustion and θ ig is the ignition angle. The ignition angle is equal to the injection angle plus the delay angle. The delay angle can be evaluated by using the Hardenberg and Hase equation: where T is the temperature at the start of injection (K), P is the pressure at the start of injection (bar), V m is the mean velocity of the piston (m/s), R is the universal gas constant, and CN is the cetane number, which can be experimentally evaluated (ASTM D613-10a Standard Test Method for Cetane Number of Diesel Fuel Oil).