1. Heat Release Modeling

Differentiate the Ideal Gas Law With Respect to Crank-Angle

Rearrange To Find Instantaneous Change in Pressure, Numerically Integrate To Find Pressure

First Law Of Thermodynamics

Rearrange to Find Instantaneous Change In Temperature, Integrate To Find Temperature

Change In Net Heat Transfer As A Function Of Crank-Angle

• The engine volume (as a function of crank angle) can be calculated using engine geometry

=clearance volume=bore=connecting rod length=crank radius (1/2 of stroke)=instantaneous distance between piston pin and crank axis

2. Cylinder Volume Modeling

3. Mass Fraction Burned Modeling

Weibe function is used to predict the combustion burn profile

=fraction of fuel mass burned at specific crank angle=Spark advance=Burn duration, = constants fit to a specific engine (approximately 5,2)

4. Heat Transfer Modeling

5. Two-Zone Model (For Burned-Zone Temperature)

burned mass→mb (i )=mb (i −1 )+d X b (i )d θ

mc

unburned mass→mu (i )=mu (i−1 )−d Xb (i )dθ

mc

unburned volume→V u (i)=( (mu ( i )V u (i −1 ) )mu (i −1 ) )( P (i )

P (i −1 ) )− 1γ u

TotalVolume→V ( i )=V b (i )+V u (i )

burned − zone temperature→Tb (i )=P ( i )V b (i)mb ( i ) R (i )

unburned −zone temperature→T u (i )=P ( i )V u (i )mu (i )R (i )

R (i )=instantaneous , fluid specific gas constant

This is found using the specific heat ratios model

ε= 44+ y

6. Atom Balancing

Species

0

0

0

Total:

KWGS=nH 2OnCO

nCO2nH 2

Water−Gas Shift Reaction→CO 2+H 2↔CO+H 2O

zeldovichmechanism→ d [NO ]dt

=2 k1f [N 2 ] e [O ]e

[N2 ]e=equilibrium concentration of N2

[O ]e=equilibrium concentration of O

k 1f=forward reaction rate coefficient

7. NO Formation Model

k 1f ( c m3

gmol− s )= (1.82∗1014 ) exp(− 38370T b )

[O ]e=kO [O2 ]e

12

(RuT b )12

Ru=UniversalGasConstant (8315[ Jkmol− K ])

K o(Pa

12 )=3.6 x 103 exp(− 31090

T b ) x (101325 )12

The equilibrium concentration of is found using the following equilibrium equation:

𝐂𝐎𝟐=𝐂𝐎+(𝟏𝟐 )𝐎𝟐

The equilibrium constant can be calculated as a function of the burned zone temperature using the JANAF tables:

𝐥𝐨𝐠𝟏𝟎𝐊𝐩=𝐊 𝐩𝐂𝐎𝟐−𝐊 𝐩𝐂𝐎 .

Use the following equation to calculate the percentage of dissociation of :

𝟏−𝛂

𝛂 (𝛂𝟐 )𝟏𝟐(𝐧𝐩

𝐏𝐩 )𝟏𝟐=𝐊𝐩

𝐧𝐩

𝐏𝐩= 𝟏 .𝟓𝐏𝐄𝐗𝐇 ( 𝐓𝐛

𝐓𝐁𝐃𝐂 )

Assume that the equilibrium mole fraction of nitrogen is equal to that provided by the atom balance equations. Assume that the composition is frozen at 90% of the peak burned-zone temperature.

8. Hydrocarbon Emissions Model (flame-quenching)

This is total crevice emissions index. A further explanation of HC formation mechanisms can be found on the Mindworks website.

The peak cylinder pressure and IMEP can be found from the single-zone model. The coolant temperature can be estimated as 350 [K]. The crevice volume can be measured.

mass of oil film→moil=ρoil π δ oil BS

ρoil=oil density 900[ kgm3 ]δ oil=oil layer thickness 3 [ μm ]

mass of HC→mHC=Pest( 1A Fmol )(

M W air

M W HC)( M W HC

M W oil H )moil

M W i=constituent molecular weights

H=Henr y ′ sConstant

8. Hydrocarbon Emissions Model (Oil Layer Absorption and Desorption

The pressure term used in predicting the mass of hydrocarbons can be estimated as an average between the inlet and peak combustion temperatures.

Pinlet (atm )=0.09875+0.00986 IMEP (kPa)

S Fwall=63024( 1IMEP ( kPa ) )( 1

A Fmol (100.0082 Toil (K ) ) B (m ) )Pest