# Derivation Of Heat Release Model Edwin Anderson's Thesis 2009 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 2. Cylinder Volume Modeling

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 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) d Xb (i) burned mass m b ( i )=m b ( i 1 ) + mc d d X b ( i) unburned mass m u ( i ) =m u ( i 1 ) mc d (

unburned volume V u ( i ) = ( m u (i ) V u ( i 1 ) ) m u ( i 1 ) )( P (i) P ( i 1 ) Total Volume V ( i )=V b (i ) +V u ( i ) )

1 u burned zone temperature T b ( i )= P (i )V b(i ) m b ( 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

Thisis found using the specific heat ratios model 6. Atom Balancing = 4 4 + y Species 0

0 0 0 0 0 Total: Water Gas Shift Reaction C O2 +H 2 CO+ H 2 O K WGS = nH 2

O nCO n CO2 n H 2 7. NO Formation Model d [ NO ] zeldovich mechanism =2 k 1 f [ N 2 ] e [ O ] e

dt [ N 2 ]e =equilibrium concentration of N 2 [ O ] e =equilibrium concentration of O k 1 f =forward reaction rate coefficient k1f ( 3 cm 38370 14

( ) = 1.82 10 exp gmol s Tb ) ( [ O ]e= ) 1 2

2 e 1 2 b kO [ O ] ( RuT ) ( Ru =Universal Gas Constant 8315 Ko

( 1 2 ) [ J kmol K ]) 1

31090 P a = 3.6 x 1 0 exp x (101325 ) 2 Tb 3 ( ) The equilibrium concentration of is found using the following equilibrium equation: =+

( ) The equilibrium constant can be calculated as a function of the burned zone temperature using the JANAF tables: = . Use the following equation to calculate the percentage of dissociation of :

= ( ) ( ) . =

( ) 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. 8. Hydrocarbon Emissions Model (Oil Layer Absorption and Desorption mass of oil film moil =oil oil BS oil =oil density 900 kg 3 m [ ]

oil =oil layer thickness 3 [ m ] mass of HC m HC =P est ( 1 A F mol )( M W air M W HC )( M W HC

m oil M W oil H M W i=constituent molecular weights H= Henr y sConstant ) 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 F wall =63024

( 1 IMEP ( kPa ) )( A F 1 mol ( 10 0.0082 T oil ( K ) ) B (m)

) P est