Simple Finite Heat Release Model (SI Engine) Introduction In the following, a finite burn duration is taken into account, in which combustion occurs at θsoc (Start Of Combustion), and continues until θeoc (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 Qin is delivered to the cylinder over a finite crank angle duration. Qin can be estimated considering the expression: Qin = mc×k = ma/α×k = (C×ρ×λv)/α×k where: mc = mass of combustible ma = mass of air C = single cylinder capacity ρ = air density (at intake pressure - 1.2 kg/m3 at 1 bar) λv = volumetric efficiency α = air-fuel ratio (gasoline stoichiometric air-fuel ratio 14.65) k = lower heating value (for gasoline 44000 kJ/kg)

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 = mcvdT, results in:

The ideal gas equation is PV = mRT, so

and

θsoc θeoc

The first law now becomes

Further reducing the equation:

Using R = cp – cv and k = cp/cv, 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=Vcc+Apist × s where Vcc is the volume of the combustion chamber, Apist 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 = Apist × ds/dθ For heat release term, ∂Q/dθ, the Wiebe function for the burn fraction is used.

Where:

f = the fraction of heat added θ = the crank angle θ0= θsoc = angle of the start of the heat addition (Start Of Combustion) ∆θ = θeoc – θsoc = 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 Qin is the overall heat input.

m

Taking the derivative of the heat release function, f, with respect to crank angle, gives the following definition of df/dθ.

If θsoc ≤ θ ≤ θeoc, 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(θn) = P(θn-1) + dP/dθ|n × (θn – θn-1).

Figure 2: Wiebe function

Figure 3: Derivative of the Wiebe function

m – 1 ma

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 PVkc = PscVsc

kc = Const. (typical exponent kc= 1.35). Psc and Vsc are pressure and volume at the beginning of the compression phase, respectively. 3. The expansion phase is modeled considering a general polytropic transformation PVke = PseVse

ke = Const. (typical exponent ke = 1.25). Pse and Vse 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 xb can be written as

where xp is the mass fraction of fuel burnt in the premixed combustion period, xdi is the mass fraction of fuel burnt in the diffusive combustion period, ∆θp and ∆θdi are, respectively, the duration of premixed and diffusive combustion and θig 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), Vm 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).