MAE 5310: COMBUSTION FUNDAMENTALS
Coupled Thermodynamic and Chemical Systems:
Constant Pressure and Constant Volume Reactors
October 22, 2012
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
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MOTIVATION
•
•
Calculation of flame temperature given only initial and final states as determined
by equilibrium, but no requirements on chemical rates (Chapter 1)
Development of chemical rate equations and chemical time scales (Chapter 2)
•
Couple chemical kinetics with fundamental conservation principles (mass and
momentum) for 4 archetypal thermodynamic systems
1. Constant-pressure, fixed mass reactor
2. Constant-volume, fixed-reactor
3. Well-Stirred reactor
4. Plug-flow reactor
•
Coupling allows description of detailed evolution of a reacting system from its
initial reactant state to final product state
– System may or may not be in chemical equilibrium
•
Goal: Calculate the system temperature and various species concentrations as
functions of time as system proceeds from reactants to products
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4 USEFUL REACTOR MODELS
1
2
3
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SUMMARY OF USEFUL RELATIONS
 MW

Mole / mass fraction relation
mix

 i  Yi 
MW
i 

X i   PMWmixYi

RTMW i
Yi 
X i MWi
 X MW
j
Yi 
MWi
i: mole fraction
Yi: mass fraction
[Xi]: molar concentration
Mass fraction / molar concentration
j
j
X i    i
i 
P

 i
RT
MWmix
X i 
 X 
Mole fraction / molar concentration
j
j
 i  Yi  X i MWi
MWmix 
Mass concentration
1
Y
i MWi
i
MWmix defined in terms of mass fractions
MWmix    i MWi
i
 X MW

 X 
i
MWmix
i
MWmix defined in terms of mole fractions
i
i
i
MWmix defined in terms of molar concentrations
1. CONSTANT PRESSURE, FIXED MASS REACTOR
dh
Q  m
dt
1st Law
 dh  
dh 1   dN 
   hi
    N i i 
dt m  i  dt  i 
dt 
Differentiation of enthalpy
Note that enthalpy’s are on per mole basis
dhi hi dT
dT

 c p ,i
dt T dt
dt
Calorically perfect gas
N i  V X i 
dN i
 V i
dt
 i short hand notation for net production rate for
 Q 
    hi i
dT  V  i

dt
 X i c pi
Substitution into 1st Law
complete mechanism


i
m
V
 X i MWi 
Volume expression
i
   i

d X i 
1 dT 

i
  i   X i 

 X  T dt 
dt

i

 i
Expression for rate of change of molar concentrations
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2. CONSTANT VOLUME, FIXED MASS REACTOR
du

Qm
dt
1st Law
 Q 
    ui i
dT  V  i

dt
 X i cvi

Substitution into 1st Law

i
 Q 
   RT   i   hi i
dT  V 
i
i

dt
  X i  c pi  R
 

In terms of molar enthalpy’s (instead of
internal energy)
i
dP
dT

 RT  i  R   X i 
dt
dt
i
i
Expression for time rate of change of pressure
Very useful for explosion calculations
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EXAMPLE: ENGINE KNOCK (LECTURE 1)
Flame Mode
•
•
Non-Flame Mode
•
•
In internal combustion engines,
compressed gasoline-air mixtures have a
tendency to ignite prematurely rather than
burning smoothly
This creates engine knock, a characteristic
rattling or pinging sound in one or more
cylinders
Octane number of gasoline is a measure
of its resistance to knock (or its ability to
wait for a spark to initiate a flame).
Octane number is determined by
comparing the characteristics of a
gasoline to isooctane (2,2,4trimethylpentane) and heptane.
– Isooctane is assigned an octane
number of 100. It is a highly
branched compound that burns
smoothly, with little knock.
– Heptane is given an octane rating of
zero. It is an unbranched compound
and knocks badly.
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EXAMPLE: ENGINE KNOCK
•
•
•
In spark ignition engines, knock occurs when unburned fuel-air mixture ahead of flame reacts
homogeneously, i.e., it autoignites
Rate of pressure rise is a key parameter in determining knock intensity and propensity for mechanical
damage to piston-crank engine assembly
Pressure vs. time traces for normal and knocking combustion in a spark-ignition engine shown below
– Note very rapid pressure rise in case of heavy knock.
Piston exposed to long
terms effects of knock
http://www-cms.llnl.gov/s-t/int_combustion_eng.html
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•
•
•
EXAMPLE: ENGINE KNOCK
Create a simple constant volume model of autoignition process and determine temperature, pressure and fuel
and product concentrations as a function of time
Assume that initial conditions corresponding to compression of a fuel-air mixture from 300 K and 1 atm to
TDC for a compression ratio of 10:1. Initial volume before compression is 3.68x10 -4 m3 which corresponds to
an engine with both bore and a stroke of 75 mm. Use ethane, C2H6, as fuel.
Other assumptions:
1. One-step global kinetics using rate parameters for ethane
2. Fuel, air and products all have equal molecular weights, MW=29
3. Specific heats for the fuel, air, and products are constant and equal, cp=1,200 J/kg K
4. Enthalpy of formation of air and products is zero and enthalpy of formation of fuel is 4x10 7 J/kg
5. Stoichiometric air-fuel ratio is 16, and combustion is restricted to stoichiometric or lean cases
y  k globa l
y

C x H y   x  O2  xCO2  H 2O
4
2

d Cx H y
E
 C H m O n
  A exp  a
2
RT  x y

dt




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SOLUTION: MATLAB SIMULATION, CONSTANT VOLUME
Products
Oxidizer
Fuel
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SOLUTION: EXPANDED SCALE ON TOP PLOT
Oxidizer
Products
Fuel
Large temperature increase in ~ 0.1 ms
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EXAMPLE RESULTS AND COMMENTS
• Equations are integrated numerically using MATLAB
• Coupled ODE’s are stiff
• Temperature increases only about 200 K in first 3 ms, then T rises extremely
rapidly to adiabatic flame temperature, Tad ~ 3300 K, in less than 0.1 ms
• This rapid temperature rise and rapid consumption of fuel is characteristic of a
thermal explosion, where the energy released and temperature rise from reaction
feeds back to produce ever-increasing reaction rates because of the (-Ea/RT)
temperature dependence of the reaction rate.
• It can also be shown that huge pressure derivatives are associated with exploding
stage of reaction, with peak values of dP/dt ~ 1.9x1013 Pa/s !!!
• Although this model predicted explosive combustion of mixture after an initial
period of slow combustion, as is observed in real knocking combustion, single-step
kinetics mechanism does not model true behavior of autoigniting mixtures
– In reality, induction period, or ignition delay, is controlled by formation of
intermediate species (radicals)
– To accurately model knock, a more detailed mechanism would be required
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HOMEWORK #3: PART 1
•
•
•
•
•
Explicitly derive all relevant equations and initial conditions shown in class for the constant
volume engine knock simulation
Calculate the actual value of the specific heat ratio, gT, for ethane C2H6
Use an program to solve the set of coupled differential equations
If possible make your code ‘intelligent’ using a variable time step based on some
convergence criteria related to temperature gradient
– Generate plots of fuel, oxidizer and product molar concentrations versus time
– Generate a plot of temperature versus time
– Generate a plot of dP/dt versus time
– Repeat for f = 0.7 and comment on results
– Repeat with methane fuel, CH4 with f=1.0 and f=0.7 and comment on results
Discuss the following issues in detail:
– How would you modify your code to account for variable molecular weights and
specific heats, i.e. which governing ODE’s change and how?
– How would you update your code to utilize the 4-step quasi-global mechanism on page
156?
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