Model Calculations

Cylinder Charge and IVC Conditions

The SI combustion model calculations begin by determining the charge mass and composition. The flow rate into the cylinders is obtained through a speed-density like regression modified after [50]

Wcyl= α1(θevc, θivo)pim+ α0(θevc, θivo) (2.35)

where pim is the intake manifold pressure and all ai represent regression coefficients. In general,

the coefficients α1 and α0 will be functions of engine speed, valve timings, and intake temperature,

however the parameterization data contained only one engine speed and a small variation in intake temperature and so the α1 and α0 are simple quadratic functions of valve timings. The mass of

fresh air can then be calculated assuming a uniform cylinder flow rate over the duration of the intake stroke

min_a = Wcyl Neng/120Ncyl

(2.37) where Ncyl is the number of cylinders.

To calculate the total in-cylinder mas, it is also necessary to know the residual gas amount, which is obtained from a regressed through the residual gas fraction xr:= mr/mc, where mr is the

residual mass and mc is the total mass. The regression is developed manually to fit post-processed

data and takes the form

xr = a1θevc2 + a2θevc+ a3θivo+ a4pim+ a5m0f + a6θsp0 + a7 (2.38)

which has a quadratic dependency on θevc to capture the inflection in residual quantity between

rebreathing (θevc aTDC) and trapping (θevc bTDC). θivo and pim are included to represent effects

on intake air charge affecting the fraction of residal gas, and cycled delayed values for fuel mass and spark timing m0_f and θ0_sp are included to capture the affect of these inputs on exhaust temperature, which can affect the storage of residual gas. Note that the cycle delays on inputs m0_f and θ0_sp are trivial to implement as these quantities are specified by the control system and so are always known explicitly.

With mfa and xr determined, the complete charge mass can then be obtained from the equation

mc=

min_a + mf

1 − xr

(2.39) where the relation mr = xrmc has been used. The AFR can also be calculated, which we quantify

in terms of the relative to the stoichiometric air-fuel ratio λ

λ = min a mf AF Rs (2.40)

where AF Rs≈ 14.6 for gasoline. Here it is assumed that ma≡ minair, i.e. that there is no recycled

air, so that λ represents the AFR in the exhaust. In typical SI operation with near-stoichiometric mixtures and small residual amounts, this should also be very close to the AFR in the cylinder, which includes recycled air.

After the cylinder charge mass and composition have been determined, the pressure and temperature at IVC are found, starting with the pressure pivc which is obtained as a linear function

intake manifold pressure as in [46]

pivc= β1pim+ β0. (2.41)

The temperature Tivc can be calculated using the ideal gas law with the known cylinder charge

mass

Tivc =

pivcVivc

Rmc

. (2.42)

Here R ≈ 287J/kgK is the ideal gas constant for air, and the volume at IVC is calculated from the crank-slider equation V (θ) = Vcl+ πB_cyl2 4  Lcr+ ac− accos(θ) − p L2 cr− (acsin(θ))2  (2.43) where θ is the crank angle of a given event (here IVC), Vcl is the chamber clearance volume, Lcr is

connecting rod length, Bcyl is the cylinder bore, and acis one half the stroke length.

Polytropic Compression/Expansion and Constant Volume Combustion

The next step of the engine cycle is to proceed with polytropic compression and constant volume combustion. Before doing this, the crank angle of instantaneous combustion θcmb must be found. As

discussed in Assumption 3 in Sec. 2.3.1, the point of instantaneous combustion is defined through the 50% burn angle θ50 following [49],

θcmb= |θ50− θM BT50 | (2.44)

where the θ50 for max brake torque θM BT50 is taken ≡ 7◦. This implies that θ50 must be found prior

to polytropic compression, which is determined through the regression

θ50= a1θ2sp+ a2θsp+ a3mf+ a4θevc2 + a5θevc+ a6θ2ivo+ a7θivo+ a8 (2.45)

Now polytropic compression can be carried out

pbc= pivc  Vivc Vcmb nc (2.46) Tbc= Tivc  Vivc Vcmb nc−1 (2.47)

where Vcmb = V (θcmb) is the volume where combustion occurs, nc is the (constant) polytropic

exponent during compression, and the subscript “bc” indicates before combustion. At θcmb, combus-

temperature rise

Tac= Tbc+

mfQlhv

cvmc

(2.48) where Qlhv is the lower heating value for the fuel, cv is the constant volume specific heat during

combustion (taken constant), and the subscript“ac” indicates after combustion. The assumption of constant volume combustion causes the ideal gas law for the pressure after combustion to reduce to

pac= pbc

 Tac

Tbc



. (2.49)

The cylinder gasses are then expanded polytropically to EVO from the point of instantaneous combustion pevo= pac  V_cmb Vevo ne (2.50) Tevo= Tac  Vcmb Vevo nexp−1 (2.51)

where ne is the (constant) polytropic exponent during expansion.

End of Cycle Outputs

After the EVO event, the outputs from the cylinders to the air path including the exhaust temperature and flow rate are calculated, along with the cycle work. Exhaust gas blowdown is assumed to occur immediately following the EVO event at constant volume, giving a polytropic expansion down to exhaust manifold pressure

Tbd= Tevo  pem pevo 1− 1 nbd (2.52) where Tbd represents the temperature after blowdown and nbd is the (constant) polytropic exponent

during blowdown. From the blowdown temperature, the temperature of the gas flowing into the exhaust manifold is calculated using an expression based on steady-state heat convection for pipe flow put forth in Model 1 of [51]

Tem = Tw+ (Tcyl,out− Tw) exp

 h(Wcyl)A

cpWcyl



(2.53)

where the temperature at the cylinder exhaust port Tcyl,out is calculated assuming a temperature

h is a cubic regression to mass flow rate

Tcyl,out= Tbd− kWcyl (2.54)

h(Wcyl)A

cp

= a3Wcyl3 + a2Wcyl2 + a1Wcyl+ a0 (2.55)

This calculation is different from the model in [51] in that it introduces a temperature drop to the cylinder exhaust port using (2.54) and the wall temperature Tw is let vary as a coefficient in the

regression instead of being taken at atmospheric temperature.

The mass and enthalpy flow output to the exhaust manifold are computed from Wex= (mina + mf) Neng 120 Ncyl (2.56) ˙ Hex = Wexcp,exTex (2.57) cp,ex= a1Tex+ a0+ R (2.58)

where the constant volume specific heat of the exhaust gas has been approximated as a linear function of exhaust manifold temperature for the SI combustion operation regime. Note that expression (2.56) assumes that the exact amount of mass that entered the cylinder on the current cycle also leaves the cylinder during the exhaust stroke, i.e. the system is in steady-state. Another way to state this is that the residual mass is assumed to be the same on the upcoming cycle as on the current cycle.

The gross work output of the cycle is calculated noting that the compression and expansion strokes were assumed to be polytropic processes, so that the expression for work of a polytropic process can be used

Wcig = pbcVcmb− pivcVivc 1 − nc +pevoVevo− pacVcmb 1 − ne (2.59) where Wcig refers to the gross indicated cycle work. The gross indicated mean effective pressure

(IMEP) can be obtained by dividing out by the cylinder displacement volume Vd

IM EP = Wcig Vd

(2.60) and finally the net indicated mean effective pressure (NMEP) is calculated assuming a rectangular pumping loop

N M EP = IM EP − (pem− pim) (2.61)

A last detail of the SI model calculations concerns syncing the SI model with HCCI model during mode transitions. Because the HCCI model contains states for cycle-cycle couplings as will be noted

in Sec 2.4, during an SI-HCCI transition these states must be initialized by the SI combustion to be passed to the HCCI model for the first HCCI cycle. Thus, these state values are calculated during the SI/HCCI coupling cycle of an SI-HCCI transition, and are otherwise discarded. One of these states is the blowdown temperature Tbd, which is already necessary for the SI model in Eq. 2.52.

The other states correspond to the compositional variables of burned gas fraction and fuel mass fraction. The burned gas fraction bbd can be shown to reduce to an algebraic function of AFR in

steady-state, so that the cycle to cycle coupling in the following equation is not necessary bbd = mf(AF Rs+ 1) mc + xrbbd(k − 1) = AF Rs+ 1 λAF Rs+ 1 in steady-state (2.62)

The fuel mass fraction fbd is set to 0 unless the mixture is rich past a specified threshold λmin until

which it is declared that all fuel burns for simplicity in the model’s compositional relations

muf = max{0, (λmin− λ)mf}, λmin ≤ 1 (2.63)

fbd=

muf

mc

(2.64) where muf is the mass of unburnt fuel and λmin is chosen = 0.97.