MAE 5360: Hypersonic Airbreathing Engines WSR and PFR Combustor Modeling Approaches Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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 control volume Extremely useful construct to study flame stabilization, NOx formation, etc.

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

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

WSR SUMMARY Solving for temperature and species mass fraction is similar to calculation of adiabatic flame temperature Difference is that now product composition is constrained by chemical kinetics rather than by chemical equilibrium WSR (or PSR) is assumed operating at steady-state, so no time dependence Equations describing WSR are a set of coupled (T and species concentration) nonlinear algebraic equations Compared with constant pressure and constant volume reactor models which are governed by set of coupled linear, 1st order ODEs Net production rate term, although it appears to have a time derivative above it, depends only on 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

EXAMPLE 1: WSR MODELING Develop a WSR model using same simplified chemistry and thermodynamic for C2H6 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

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

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

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

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

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

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 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

EXAMPLE 2: FURTHER COMMENTS Consider again WSR model for 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

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.

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)

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

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

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

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

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