ME 475/675 Introduction to

CombustionLecture 21

Coupled chemical/thermal analysis, Constant pressure, Constant Volume, Start Ex 6.1

Announcements• HW 8, Numerical Solution to Example 6.1• Due Monday, Oct. 19, 2015

Chapter 6 Coupling Chemical and Thermal Analysis of Reacting systems• Four simple reactor systems, p 184

1. Constant pressure and fixed Mass• Time dependent, well mixed

2. Constant-volume fixed-mass• Time dependent, well mixed

3. Well-stirred reactor• Steady, different inlet and exit conditions

4. Plug-Flow• Steady, dependent on location

• Coupled Energy, species production, and state constraints • For plug flow also need momentum

since speed and pressure vary with location

Constant pressure and fixed mass reactor• Constituents • reactants and products, (book uses )• P and m constant

• Find, as a function of time, t• Temperature

• To find use conservation of energy• Molar concentration (book calls this )

• use species generation/consumption rates from chemical kinetics•

• state, mixture• Highly coupled

• Assume we know “production rates” per unit volume

• Rate depends on current molar concentration (per volume) of each constituent, and temperature• From chemical Kinetics

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First Law (EnergyConservation)

• Only boundary work:

• • Where enthalpy , and for a mixture

• Production rate ; • ; ;

• Divide by

• ; Solve for

• First order differential equation, Initial conditions (IC): • At each time step, to find the change in

• Need , and

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Change in Molar Concentrations• *• Species production and volume change affect molar concentration

• Find the volume V from ideal gas equation of state

• Take time derivative to see how volume changes with time

• Divide both sides by

• Plug into *

• Initial Conditions: at t = 0, ,

coupled System of 1st order differential equations• Initial Conditions, at t = 0

• , , and

• Assume we also know • Use the first order differentials to find and at time

• • ;

• System Volume• (algebraic not differential eqn.)

t T [1] [2] … [M] w1 w2 … wM V Q d[1]/dt d[2]/dt … d[M]/dt dT/dt

0 T0 [1]0 [2]0 … [M]0

Dt2Dt

Constant-Volume Fixed-Mass Reactor• Constant V and m • Find versus time • 1st Law

• ;

• ; so • ,

• , divide by

• , solve for • ; in contrast for constant pressure

Tabulated Data• Need to evaluate (true, but not useful)• However, tables only have

• , so use

• , so use

• (true and useful)• Initial Condition: at

• Species Production (no volume change)

Reactor Pressure• Ideal Gas Law•

• Divide by (constant)

• Pressure Rate of change (affects detonation)

Example 6.1 (p. 189) This will be HW• In spark-ignition engines, knock occurs when the unburned fuel-air mixture ahead

of the flame reacts homogeneously, i.e., it auto-ignites. The rate-of-pressure rise is a key parameter in determining knock intensity and propensity for mechanical damage to the piston-crank assembly. Pressure-versus-time traces for normal and knocking combustion in a spark-ignition engine are illustrated in Fig. 6.2. Note the rapid pressure rise in the case of heavy knock. Figure 6.3 shows schleiren (index-of-refraction gradient) photographs of flame propagation for normal and knocking combustion

•

Example 6.1• Create a simple constant-volume model of the autoignition process and determine the

temperature and the fuel and product concentration histories. Also determine the dP/dt as a function of time. Assume initial conditions corresponding to compression of a fuel-air mixture from 300 K and 1 atm to top-dead-center for a compression ratio of 10:1. The initial volume before compression is 3.68*10-4 m3, which corresponds to an engine with both a bore and a stroke of 75 mm. Use ethane as fuel. Assume:• One-step global kinetics using the rate parameters for ethane C2H6 (Table 5.1)• Fuel, air, and products all have equal molecular weights: MWF= MWOx= MWP= 29• The specific heats of the fuel, air and products are constants and equal:

• cp,F= cp,Ox= cp,Pr= 1200 J/kg K

• The enthalpy of formation of the air and products are zero, and that of the fuel is • 4*107 J/kg

• The stoichiometric air-fuel ratio is 16.0 and restrict combustion to stoichiometric or lean conditions.

Global and Quasi-global mechanisms• Empirical•

• stoichiometric mixture with not air

• • Page 157, Table 5.1: , for different fuels

• These values are based on flame speed data fit (Ch 8)• In Table 5.1 units for • However, we often want in units of

Given in Table 5.1, p. 157

Sometimes Want These Units

•