1. MAE 5310: COMBUSTION FUNDAMENTALS
Coupled Thermodynamic and Chemical Systems:
Constant Pressure and Constant Volume Reactors
March 15, 2017
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. 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
Constant-pressure, fixed mass reactor
Constant-volume, fixed-reactor
Well-Stirred reactor
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

3. 4 USEFUL REACTOR MODELS 1 2 3 4

4. SUMMARY OF USEFUL RELATIONS
Mole / mass fraction relation
Mass fraction / molar concentration
Mole fraction / molar concentration
Mass concentration
MWmix defined in terms of mass fractions
MWmix defined in terms of mole fractions
MWmix defined in terms of molar concentrations
ci: mole fraction
Yi: mass fraction
[Xi]: molar concentration

5. 1. CONSTANT PRESSURE, FIXED MASS REACTOR
1st Law
Differentiation of enthalpy
Note that enthalpy’s are on per mole basis
Calorically perfect gas
short hand notation for net production rate for complete mechanism
Substitution into 1st Law
Volume expression
Expression for rate of change of molar concentrations

6. 2. CONSTANT VOLUME, FIXED MASS REACTOR
1st Law
Substitution into 1st Law
In terms of molar enthalpy’s (instead of internal energy)
Expression for time rate of change of pressure
Very useful for explosion calculations

7. EXAMPLE: ENGINE KNOCK (LECTURE 1)
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 (2,2,4-trimethylpentane) 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.
Flame Mode
Non-Flame Mode

8. 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

9. 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:
One-step global kinetics using rate parameters for ethane
Fuel, air and products all have equal molecular weights, MW=29
Specific heats for the fuel, air, and products are constant and equal, cp=1,200 J/kg K
Enthalpy of formation of air and products is zero and enthalpy of formation of fuel is 4x107 J/kg
Stoichiometric air-fuel ratio is 16, and combustion is restricted to stoichiometric or lean cases

10. SOLUTION: MATLAB SIMULATION, CONSTANT VOLUME
Products
Oxidizer
Fuel

11. SOLUTION: EXPANDED SCALE ON TOP PLOT
Oxidizer
Products
Fuel
Large temperature increase in ~ 0.1 ms

12. 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

13. HOMEWORK #4b: 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?

14. MAE 5310: COMBUSTION FUNDAMENTALS
Coupled Thermodynamic and Chemical Systems:
Well-Stirred Reactor (WSR) Theory
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

15. WELL-STIRRED REACTOR THEORY OVERVIEW
Well-Stirred Reactor (WSR) or Perfectly-Stirred Reactor (PSR) is an ideal reactor in which perfect mixing is achieved inside the control volume
Extremely useful construct to study flame stabilization, NOx formation, etc.

16. APPLICATION OF CONSERVATION LAWS
Rate at which mass of i accumulates
within control volume
Mass flow of i into control volume
Mass flow of i out of control volume
Rate at which mass of i is
generated within control volume
Relationship between mass generation rate of a
species related to the net production rate

17. APPLICATION OF CONSERVATION LAWS
Outlet mass fraction, Yi,out is equal
to the mass fraction within the reactor
Conversion of molar concentration into mass fraction
(see slide 2)
So far, N equations with N+1 unknowns, need to close set
Application of steady-flow energy equation
Energy equation in terms
of individual species

18. WSR SUMMARY
Solving for temperature and species mass fraction is similar to calculation of adiabatic flame temperature (Glassman, Chapter 1)
The difference is that now the product composition is constrained by chemical kinetics rather than by chemical equilibrium
WSR (or PSR) is assumed to be operating at steady-state, so there is no time dependence
Compared with the constant pressure and constant volume reactor models considered previously
The equations describing the WSR are a set of coupled (T and species concentration) nonlinear algebraic equations
Compared with constant pressure and constant volume reactor models which were governed by a set of coupled linear, 1st order ODEs
Net production rate term, although it appears to have a time derivative above it, depends only on the mass fraction (or concentration) and temperature, not time
Solve this system of equations using Newton method for solution of nonlinear equations
Common to define a mean residence time, tres, for gases in WSR

19. EXAMPLE 1: WSR MODELING
Develop a WSR model using same simplified chemistry and thermodynamic used in previous example
Equal constant cp’s, MW’s, one-step global kinetics for C2H6
Use model to develop blowout characteristics of a spherical reactor with premixed reactants (C2H6 and Air) entering at 298 K. Diameter of reactor is 80 mm.
Plot f at blowout as a function of mass flow rate for f ≤ 1.0 and assume that reactor is adiabatic
Set of 4 coupled nonlinear algebraic equations with unknowns, YF, YOx, YPr, and T
Treat mass flow rate and volume as known parameters
To determine reactor blowout characteristic, solve nonlinear algebraic equations on previous slide for a sufficiently small value of mass flow rate that allows combustion at given equivalence ratio
Increase mass flow rate until failure to achieve a solution or until solution yields input values

20. EXAMPLE 1: RESULTS AND COMMENTS
Decreasing conversion of fuel to products as mass flow rate is increased to blowout condition
Decreased temperature as flow rate is increased to blowout condition
Mass flow rate for blowout is about kg/s
Ratio of blowout temperature to adiabatic flame temperature is 1738 / 2381 = 0.73
Repeat calculations at various equivalence ratios generates the blowout characteristic curve
Reactor is more easily blown out as the fuel-air mixture becomes leaner
Shape of blowout curve is similar to experimental for gas turbine engine combustors

21. EXAMPLE 2: GAS TURBINE COMBUSTOR CHALLENGES
Based on material limits of turbine (Tt4), combustors must operate below stoichiometric values
For most relevant hydrocarbon fuels, ys ~ 0.06 (based on mass)
Comparison of actual fuel-to-air and stoichiometric ratio is called equivalence ratio
Equivalence ratio = f = y/ystoich
For most modern aircraft f ~ 0.3

22. EXAMPLE 2: WHY IS THIS RELEVANT?
Most mixtures will NOT burn so far away from stoichiometric
Often called Flammability Limit
Highly pressure dependent
Increased pressure, increased flammability limit
Requirements for combustion, roughly f > 0.8
Gas turbine can NOT operate at (or even near) stoichiometric levels
Temperatures (adiabatic flame temperatures) associated with stoichiometric combustion are way too hot for turbine
Fixed Tt4 implies roughly f < 0.5
What do we do?
Burn (keep combustion going) near f=1 with some of ingested air
Then mix very hot gases with remaining air to lower temperature for turbine

23. SOLUTION: BURNING REGIONS
Turbine
Air
Primary
Zone
f~0.3
f ~ 1.0
T>2000 K
Compressor

24. COMBUSTOR ZONES: MORE DETAILS
Primary Zone
Anchors Flame
Provides sufficient time, mixing, temperature for “complete” oxidation of fuel
Equivalence ratio near f=1
Intermediate (Secondary Zone)
Low altitude operation (higher pressures in combustor)
Recover dissociation losses (primarily CO → CO2) and Soot Oxidation
Complete burning of anything left over from primary due to poor mixing
High altitude operation (lower pressures in combustor)
Low pressure implies slower rate of reaction in primary zone
Serves basically as an extension of primary zone (increased tres)
L/D ~ 0.7
Dilution Zone (critical to durability of turbine)
Mix in air to lower temperature to acceptable value for turbine
Tailor temperature profile (low at root and tip, high in middle)
Uses about 20-40% of total ingested core mass flow
L/D ~

25. EXAMPLE 2: GAS TURBINE ENGINE COMBUSTOR
Consider primary combustion zone of a gas turbine as a well-stirred reactor with volume of 900 cm3. Kerosene (C12H24) and stoichiometric air at 298 K flow into the reactor, which is operating at 10 atm and 2,000 K
The following assumptions may be employed to simplify the problem
Neglect dissociation and assume that the system is operating adiabatically
LHV of fuel is 42,500 KJ/kg
Use one-step global kinetics, which is of the following form
Ea is 30,000 cal/mol = 125,600 J/mol
Concentrations in units of mol/cm3
Find fractional amount of fuel burned, h
Find fuel flow rate
Find residence time inside reactor, tres

26. EXAMPLE 2: FURTHER COMMENTS
Consider again the WSR model for the gas turbine combustor primary zone, however now treat temperature T as a variable.
At low T, fuel mass flow rate and h are low
At high T, h is close to unity but fuel mass flow rate is low because the concentration [F] is low ([F]=cFP/RT), which reduces reaction rate
In the limit of h=1, T=Tflame and the fuel mass flow rate approaches zero
For a given fuel flow rate two temperature solutions are possible with two different heat outputs are possible
f=1, kerosene-air mixture
V=900 cm3
P=10 atm

27. EXAMPLE #3: HOW CHEMKIN WORKS
Detailed mechanism for H2 combustion
Reactor is adiabatic, operates at 1 atm, f=1.0, and V=67.4 cm3
For residence time, tres, between equilibrium and blow-out limits, plot T, cH2O, cH2, cOH, cO2, cO, and cNO vs tres.

28. EXAMPLE #3: HOW CHEMKIN WORKS
Tflame and cH2O concentration drop as tres becomes shorter
H2 and O2 concentrations rise
Behavior of OH and O radicals is more complicated
NO concentration falls rapidly as tres falls below 10-2 s
Input quantities in CHEMKIN:
Chemical mechanism
Reactant stream constituents
Equivalence ratio
Inlet temperature and pressure
Reactor volume
tres (1 ms ~ essentially equilibrated conditions)

29. MAE 5310: COMBUSTION FUNDAMENTALS
Coupled Thermodynamic and Chemical Systems:
Plug Flow Reactor
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

30. PLUG FLOW REACTOR OVERVIEW
T=T(x)
[Xi]=[Xi](x)
P=P(x)
V=u(x) Dx
Assumptions
Steady-state, steady flow
No mixing in the axial direction. This implies that molecular and/or turbulent mass diffusion is negligible in the flow direction
Uniform properties in the direction perpendicular to the flow (flow is one dimensional). This implies that at any cross-section, a single velocity, temperature, composition, etc., completely characterize the flow
Ideal frictionless flow. This assumption allows the use Euler equation to relate pressure and velocity
Ideal gas behavior. State relations to relate T, P, r, Yi, and h
Goal: Develop a system of 1st order ODEs whose solution describes the reactor flow properties, including composition, as a function of distance, x

31. GOVERNING EQUATIONS
Mass conservation
x-momentum conservation
Energy conservation
P is the local perimeter of the reactor
Species conservation

32. USEFUL FORMS Results from expanding conservation of mass
Results from expanding the energy equation
Differentiation of functional relationship for ideal-gas calorific equation of state, h=h(T,Yi)
Differentiation of ideal-gas equation of state
Differentiation of definition of mixture molecular weight expressed in terms of species mass fractions

33. POTENTIAL SOLUTION SET
In these equations the heat transfer rate has been set to zero for simplicity
Mathematical description of the plug-flow reactor is similar to constant pressure and constant volume reactor models developed previously
All 3 result in a coupled set of ODEs
Plug Flow Reactor are expressed as functions of spatial coordinate, x, rather than time, t

34. APPLICATION TO COMBUSTION SYSTEM MODELING
Turbine
Air
Primary
Zone
f~0.3
f ~ 1.0
T~2500 K
Compressor
Conceptual model of a gas-turbine combustor using 2 WSRs and 1 PFR