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MAE 5310: COMBUSTION FUNDAMENTALS

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. MOTIVATION.

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MAE 5310: COMBUSTION FUNDAMENTALS

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  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 4 3

  4. SUMMARY OF USEFUL RELATIONS ci: mole fraction Yi: mass fraction [Xi]: molar concentration 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

  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 http://www-cms.llnl.gov/s-t/int_combustion_eng.html

  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 0.193 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 ~ 1.5-1.8

  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 • 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 T=T(x) [Xi]=[Xi](x) P=P(x) V=u(x) Dx

  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

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