ME 557: Combustion 1

Sources and References

• Turns, S. R. An Introduction to Combustion: Concepts and Applications, 3rd edition, McGraw-Hill, 2012.
• Glassman, I., Yetter, R. A., and Glumac, N. G. Combustion, 5th edition, Academic Press, 2015.
• Kuo, K. K. Principles of Combustion, 2nd edition, Wiley, 2005.
• Law, C. K. Combustion Physics, Cambridge University Press, 2006.
• Warnatz, J., Maas, U., and Dibble, R. W. Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, 4th edition, Springer, 2006.
• Lefebvre, A. H., and Ballal, D. R. Gas Turbine Combustion: Alternative Fuels and Emissions, 3rd edition, CRC Press, 2010.
• Williams, F. A. Combustion Theory, 2nd edition, Benjamin/Cummings, 1985.
• Peters, N. Turbulent Combustion, Cambridge University Press, 2000.
• Poinsot, T., and Veynante, D. Theoretical and Numerical Combustion, 3rd edition, 2012.
• Lieuwen, T. C. Unsteady Combustor Physics, Cambridge University Press, 2012.
• MIT OpenCourseWare 2.650 / 10.575, Introduction to Sustainable Energy and combustion units.
• Stanford University ME 370B Fundamentals of Combustion, publicly available lecture materials.
• Smith, G. P. et al. GRI-Mech 3.0, www.me.berkeley.edu/gri_mech, 1999.
• Gordon, S., and McBride, B. J. Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications (CEA), NASA RP-1311, 1994.
• Goodwin, D. G. et al. Cantera: An Object-Oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes, cantera.org.

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Course Overview

Combustion is the coupling of chemistry and fluid mechanics at temperatures hot enough for the chemistry to release significant heat and restructure the flow. ME 557 treats this coupling at the level of a first graduate course: enough thermodynamics and kinetics to predict what a flame can do in principle, enough transport theory to predict how fast it does it, and enough structural analysis of canonical flames, premixed and diffusion, laminar and turbulent, to reason about real devices such as burners, engines, gas turbines, and furnaces.

The course is built around a small number of physical ideas that recur at every scale. Stoichiometry and chemical equilibrium fix end states. Arrhenius kinetics and chain-branching set timescales. A reaction–diffusion balance between a thin reaction zone and an upstream preheat zone sets laminar flame speed. The same balance, viewed through characteristic ratios of chemical and fluid timescales, classifies turbulent flames. In non-premixed systems, a conserved scalar (mixture fraction) replaces reaction progress as the natural coordinate, and the flame sits wherever that scalar equals its stoichiometric value. Ignition and extinction are transitions in a thermal balance between heat release and heat loss. Detonation is the shock-coupled limit of flame propagation.

These notes synthesize the standard graduate references listed at the end. Nothing here is tied to a specific offering of the course.

1. Combustion Thermodynamics

1.1 Stoichiometry, Equivalence Ratio, and Mixture Composition

A global combustion reaction for a generic hydrocarbon fuel burning in air is written

\[ \mathrm{C}_x\mathrm{H}_y + \left( x + \tfrac{y}{4} \right) \left( \mathrm{O}_2 + 3.76\,\mathrm{N}_2 \right) \rightarrow x\,\mathrm{CO}_2 + \tfrac{y}{2}\,\mathrm{H}_2\mathrm{O} + 3.76\left( x + \tfrac{y}{4} \right)\mathrm{N}_2 \]

The 3.76 is the molar ratio of nitrogen to oxygen in dry air. The stoichiometric fuel–air mass ratio \( \left( F / A \right)_s \) is set by atomic balance and the molecular weights. Actual mixtures are characterized by the equivalence ratio

\[ \phi = \frac{\left( F / A \right)}{\left( F / A \right)_s} \]

Lean mixtures have \( \phi < 1 \) and leave excess oxygen; rich mixtures have \( \phi > 1 \) and leave excess fuel, typically appearing as CO and H_2 rather than unreacted hydrocarbons. An alternative, and in many modern treatments preferable, coordinate is the mixture fraction

\[ Z = \frac{s\,Y_F - Y_O + Y_{O,2}}{s\,Y_{F,1} + Y_{O,2}} \]

where \( Y_F \) and \( Y_O \) are fuel and oxidizer mass fractions, subscript 1 denotes the fuel stream, subscript 2 denotes the oxidizer stream, and \( s \) is the mass of oxidizer consumed per unit mass of fuel. \( Z \) is a conserved scalar when fuel and oxidizer have equal diffusivities, so it is unchanged by reaction and obeys a pure convection–diffusion equation. The stoichiometric surface \( Z = Z_{st} \) is where a diffusion flame sits.

1.2 Heats of Reaction and Heating Values

The standard heat of reaction at temperature \( T \) is the enthalpy difference between products and reactants at that temperature with each species in its reference state. The lower heating value (LHV) takes product water as vapor; the higher heating value (HHV) takes it as liquid. The difference is the latent heat of vaporization of the water produced. Practical device efficiencies almost always quote LHV because exhaust water leaves as vapor.

For an element in its stable form at 298.15 K the enthalpy of formation is zero by convention. For a compound, \( \Delta h_f^\circ \) is the enthalpy of formation from the elements at that reference state. Species enthalpy at arbitrary \( T \) is

\[ h_i\!\left( T \right) = \Delta h_{f,i}^\circ + \int_{298.15}^{T} c_{p,i}\!\left( T' \right)\,dT' \]

Polynomial fits (NASA nine-coefficient or Shomate forms) give \( c_p\!\left( T \right) \) accurately over wide temperature ranges and are the workhorse of any combustion calculation.

1.3 Adiabatic Flame Temperature

For a constant-pressure adiabatic reactor with negligible kinetic energy changes,

\[ \sum_{reactants} n_i h_i\!\left( T_r \right) = \sum_{products} n_j h_j\!\left( T_{ad} \right) \]

If products are fixed by complete combustion (no dissociation), \( T_{ad} \) is the solution of a single nonlinear equation in \( T_{ad} \). For stoichiometric hydrocarbon–air mixtures this gives \( T_{ad} \) in the range of roughly 2200 to 2400 K. Constant-volume adiabatic temperatures are a few hundred kelvin higher because no pdV work is done.

Complete-combustion \( T_{ad} \) overestimates the true value above about 1800 K because dissociation of CO_2 into CO + (1/2) O_2, of H_2O into OH and H_2, and of N_2 and O_2 into O, N, NO, and so on absorbs energy. Accurate values require equilibrium composition at \( T_{ad} \), solved by minimizing Gibbs free energy.

1.4 Chemical Equilibrium

For a reversible reaction written as \( \sum_i \nu_i' A_i \rightleftharpoons \sum_i \nu_i'' A_i \), equilibrium at temperature \( T \) and pressure \( p \) satisfies

\[ K_p\!\left( T \right) = \prod_i \left( \frac{p_i}{p^\circ} \right)^{\nu_i'' - \nu_i'} = \exp\!\left( -\frac{\Delta G^\circ\!\left( T \right)}{R T} \right) \]

where \( \Delta G^\circ \) is the change in standard Gibbs free energy of reaction. In practice one solves a coupled set: element conservation, mass balance, and one \( K_p \) equation per independent reaction. For hydrocarbon–air flames a ten- to twelve-species product set (CO_2, H_2O, CO, H_2, O_2, N_2, OH, H, O, NO, N, possibly NO_2) is sufficient.

An equivalent formulation minimizes total Gibbs energy \( G = \sum_i n_i \mu_i \) subject to atom conservation using Lagrange multipliers. The Gordon–McBride CEA code and cantera are built on this method. Results give the equilibrium temperature, pressure, and composition for adiabatic and non-adiabatic processes.

2. Chemical Kinetics of Combustion

2.1 Elementary Reactions and the Law of Mass Action

An elementary reaction proceeds as written, molecule by molecule, with its rate given by the law of mass action. For \( A + B \rightarrow C + D \),

\[ \frac{d\!\left[ C \right]}{dt} = k\!\left( T \right)\,\left[ A \right]\left[ B \right] \]

The rate coefficient is usually expressed in modified Arrhenius form,

\[ k\!\left( T \right) = A\,T^{n}\,\exp\!\left( -\frac{E_a}{R T} \right) \]

with pre-exponential factor \( A \), temperature exponent \( n \), and activation energy \( E_a \). The exponential makes the rate extraordinarily sensitive to temperature; at a flame temperature of 2000 K, an activation energy of 200 kJ/mol gives a rate that changes by a factor of two for each 50 K change in temperature. This is the mathematical origin of the sharpness of flames.

Termolecular reactions, such as radical recombination A + B + M → AB + M, depend on a third body M that carries off the excess energy. Their rates depend on total pressure and on the identity of M through efficiency factors.

2.2 Steady-State and Partial-Equilibrium Approximations

A detailed mechanism (GRI-Mech 3.0 for methane has 53 species and 325 reactions, USC Mech II for C1–C4 has nearly 800 reactions) is too large for hand analysis. Two approximations help. The quasi-steady-state approximation (QSSA) sets \( d\!\left[ X \right] / dt \approx 0 \) for a short-lived intermediate, so its production and consumption rates balance, giving an algebraic relation between it and longer-lived species. The partial-equilibrium approximation (PEA) assumes that a fast reaction is effectively reversible, so its forward and backward rates balance. Combining QSSA and PEA reduces detailed mechanisms to a few global steps.

2.3 Chain Reactions and Branching

Most combustion proceeds by radical chains. An initiation step creates radicals from stable molecules at high temperature. Propagation steps consume one radical and produce another. Branching steps consume one radical and produce more than one. Termination removes radicals. For the canonical H_2 oxidation,

\[ \mathrm{H} + \mathrm{O}_2 \rightarrow \mathrm{OH} + \mathrm{O} \]\[ \mathrm{O} + \mathrm{H}_2 \rightarrow \mathrm{OH} + \mathrm{H} \]\[ \mathrm{OH} + \mathrm{H}_2 \rightarrow \mathrm{H}_2\mathrm{O} + \mathrm{H} \]

The first step is chain-branching: each H atom consumed produces two new radicals (OH and O). Branching makes the radical pool grow exponentially until fuel is consumed. The competition between branching and termination sets explosion limits, the famous Z-shaped pressure–temperature curve for hydrogen–oxygen.

2.4 Hydrogen and Hydrocarbon Oxidation

Hydrogen oxidation is the foundation: its mechanism underlies every hydrocarbon mechanism because the same H, O, OH, HO_2, H_2O_2 chemistry handles the last steps of any carbon oxidation. Key reactions include the branching H + O_2 → OH + O, the three-body H + O_2 + M → HO_2 + M that dominates at high pressure, and HO_2 chemistry controlling low-temperature behavior.

Carbon monoxide oxidation is notoriously slow in dry systems: CO + O_2 has a huge barrier. Real CO oxidation proceeds via CO + OH → CO_2 + H, so it requires hydrogen-bearing species. This is why CO accumulates in fuel-rich or cold regions and why CO emissions correlate with poor mixing.

Hydrocarbon oxidation proceeds through hierarchical steps. A fuel molecule undergoes H-abstraction by OH, H, O, or HO_2. The resulting alkyl radical decomposes by β-scission to smaller alkenes and alkyl radicals. At high temperature, oxidation ultimately produces CO that is then oxidized to CO_2. At low-to-intermediate temperatures (below about 1000 K), alkyl + O_2 addition and internal isomerization lead to peroxide chemistry responsible for cool flames, two-stage ignition, and engine knock. The negative temperature coefficient region, where ignition delay lengthens with increasing temperature, is a signature of this low-temperature branch.

3. Fuel Properties and Flammability

3.1 Heating Values, Flammability Limits, and Stoichiometric Air

Table 1 summarizes representative properties of common fuels. LHV is on a mass basis; flammability limits are volumetric, in air at 1 atm and 298 K.

Fuel	LHV (MJ/kg)	\( \phi \) at LFL	\( \phi \) at UFL	Autoignition (K)
Methane	50.0	0.50	1.70	810
Propane	46.3	0.51	2.83	740
n-Heptane	44.6	0.53	4.50	495
iso-Octane	44.3	0.55	4.00	690
Hydrogen	120.0	0.10	7.17	845
Carbon monoxide	10.1	0.34	6.76	880
Methanol	19.9	0.48	4.08	700
Jet-A	43.2	0.60	3.00	500

Hydrogen stands out for its extreme flammability range and high mass-based heating value, but its low volumetric energy density and wide range also make it a safety challenge. Heavy hydrocarbons have narrower lean limits and lower autoignition temperatures because of low-temperature radical chemistry.

3.2 Minimum Ignition Energy

The minimum ignition energy (MIE) is the smallest spark energy that establishes a self-propagating flame. For hydrocarbon–air mixtures at 1 atm, MIE is typically 0.2 to 0.3 mJ near stoichiometric; for hydrogen–air it drops to about 0.02 mJ. MIE scales approximately with

\[ E_{min} \sim \rho\,c_p\,\left( T_{ad} - T_u \right)\,\delta_{q}^{3} \]

where \( \delta_q \) is the quenching distance and \( T_u \) the unburned temperature. The cube on quenching distance makes MIE extremely sensitive to mixture composition and pressure, since \( \delta_q \) falls roughly as \( 1/p \).

3.3 Autoignition Temperature and Delay

Autoignition temperature (AIT) is the lowest temperature at which a mixture self-ignites without an external source under specified conditions. Because the outcome depends on residence time and geometry, AIT is standardized (ASTM E659). The ignition delay \( \tau_{ign} \) at fixed \( T \) and \( p \) is often fit by

\[ \tau_{ign} = A\,p^{n}\,\left[ F \right]^{a}\left[ O_2 \right]^{b}\,\exp\!\left( \frac{E_a}{R T} \right) \]

with the pressure exponent \( n \) typically near \(-1\) and an effective \( E_a \) of 100 to 200 kJ/mol for high-temperature chemistry. In the NTC region for heavy hydrocarbons, \( \tau_{ign} \) first decreases with \( T \), then increases, then decreases again as chemistry shifts between low- and high-temperature branches.

4. Premixed Laminar Flames

4.1 Flame Speed and Flame Structure

A freely propagating laminar premixed flame is a one-dimensional wave of steady reaction–diffusion balance that advances into the unburned mixture at the laminar flame speed \( S_L \). The flame has two characteristic zones. The preheat zone, of thickness \( \delta_T \sim \alpha / S_L \), is where thermal and species diffusion dominate and reaction is small. The reaction zone, an order of magnitude thinner, is where exponential Arrhenius kinetics consume the fuel. Heat produced in the reaction zone diffuses upstream, raising the temperature of fresh gas until it too can react: the flame is a self-sustaining wave.

For methane–air at 1 atm, \( S_L \approx 38\),cm/s near stoichiometric and \( \delta_T \approx 0.5 \) mm. Hydrogen–air has \( S_L \) up to about 280 cm/s; heavy hydrocarbons around 40 cm/s. Flame speed peaks slightly rich of stoichiometric for hydrocarbons because diffusion of the lighter reactant dominates near the peak temperature.

4.2 Mallard–Le Chatelier Thermal Theory

The simplest predictive theory balances convective enthalpy transport with the heat conducted back from an inner flame surface at an ignition temperature \( T_i \). The energy balance gives

\[ \rho_u\,c_p\,S_L\,\left( T_i - T_u \right) = \frac{\lambda\,\left( T_b - T_i \right)}{\delta_R} \]

with \( \delta_R \) the reaction zone thickness. Using \( \delta_R \sim S_L \tau_{chem} \) (the reaction zone is swept through in a chemical time),

\[ S_L \sim \sqrt{\frac{\alpha}{\tau_{chem}}} \]

So flame speed is the geometric mean of a diffusivity and a reaction rate. This is the single most important scaling in premixed combustion.

4.3 Zel’dovich–Frank-Kamenetskii Analysis

A more rigorous asymptotic analysis exploits the large activation energy to split the flame into an inert preheat zone and a thin reaction layer. Matching the two gives

\[ S_L^2 = \frac{2\,\lambda}{\rho_u^2\,c_p}\,\frac{\left( T_b - T_u \right)}{\left( T_b - T_i \right)^2}\,\omega_R \]

where \( \omega_R \) is an integrated reaction rate across the reaction zone. Taking \( \omega_R \sim \rho\,A\,\exp\!\left( -E_a / R T_b \right) \) reproduces the Mallard–Le Chatelier scaling but with the correct \( T_b \)-dependence. The activation temperature \( T_a = E_a / R \) sets the Zel’dovich number \( Ze = E_a\,\left( T_b - T_u \right)/\left( R T_b^2 \right) \). Large \( Ze \) justifies the thin reaction zone and makes the flame extremely sensitive to changes in \( T_b \). A 50 K drop in \( T_b \) can halve \( S_L \).

4.4 Pressure and Temperature Dependence

Empirically \( S_L \propto T_u^{\alpha} p^{\beta} \) with \( \alpha \) near 1.5 to 2 and \( \beta \) near \(-0.25\) to \(-0.5\) for methane–air, reflecting that increasing pressure pushes chemistry toward three-body termination (H + O_2 + M → HO_2 + M) that competes with the main branching step.

5. Flame Stretch, Stability, and the Lewis Number

5.1 Stretch Rate

A real flame is rarely planar. Curvature and hydrodynamic strain deform it. The total stretch rate is

\[ \kappa = \frac{1}{A}\,\frac{dA}{dt} \]

where \( A \) is an element of flame surface area. Stretch has two components: curvature contributes \( S_d / R \) with \( R \) a local radius of curvature, and tangential strain contributes the projected gradient of the tangential velocity.

5.2 Markstein Length and Lewis Number Effects

The stretched flame speed is related to the unstretched one by

\[ S_L\!\left( \kappa \right) = S_L^{0} - L_M\,\kappa \]

for small stretch, where \( L_M \) is the Markstein length. Its sign is controlled by the mixture Lewis number

\[ Le = \frac{\alpha}{D_F} \]

the ratio of thermal to deficient-reactant mass diffusivity. When \( Le < 1 \), heat diffuses more slowly than fuel, and in a convex-toward-unburned bulge, fuel diffuses in faster than heat diffuses out, locally enriching and heating, so stretch accelerates the flame. When \( Le > 1 \), stretch slows it. This is the physical origin of the diffusional–thermal instability responsible for the cellular flames observed in lean H_2–air and rich hydrocarbon–air, and for the smoothing of rich H_2–air and lean hydrocarbon–air flames.

Hydrodynamic (Darrieus–Landau) instability arises from the density jump across the flame itself: a planar flame front is unconditionally unstable to long-wavelength disturbances regardless of chemistry. In practice this instability is masked or moderated by curvature effects and confinement but reappears in large flames, leading to the wrinkled fronts seen in spherical bomb experiments.

5.3 Extinction by Stretch

Push stretch too far and the flame extinguishes. The critical extinction stretch \( \kappa_{ext} \) scales as \( S_L^2 / \alpha \), the inverse of a flame residence time. Counterflow burner experiments measure \( \kappa_{ext} \) directly and provide benchmark data for mechanism validation.

6. Turbulent Premixed Combustion

6.1 Non-dimensional Groups

Turbulent premixed flames are classified by ratios of flow and chemical scales. The turbulent Reynolds number compares turbulent transport to molecular transport:

\[ Re_t = \frac{u'\,\ell_t}{\nu} \]

with \( u' \) the rms turbulent velocity and \( \ell_t \) the integral length scale. The Damköhler number is the ratio of a flow time to a chemical time,

\[ Da = \frac{\tau_t}{\tau_c} = \frac{\ell_t / u'}{\delta_L / S_L} \]

The Karlovitz number compares chemical time to Kolmogorov time,

\[ Ka = \frac{\tau_c}{\tau_\eta} = \frac{\delta_L^2}{\eta^2} = \left( \frac{u_\eta}{S_L} \right)^2 \]

These three groups are related by \( Re_t = Da^2 Ka^2 \).

6.2 The Borghi–Peters Regime Diagram

Plotting \( u' / S_L \) against \( \ell_t / \delta_L \) on log axes produces the regime diagram. Laminar flames sit at low \( Re_t \). Wrinkled and corrugated flamelets occupy \( Ka < 1 \): turbulence wrinkles the flame but does not disturb its internal structure. Crossing \( Ka = 1 \) (the Klimov–Williams criterion) the smallest turbulent eddies penetrate the flame, broadening the reaction zone. At \( Ka > 100 \) or so, sometimes called the well-stirred reactor limit, chemistry and turbulence are fully intermixed.

Engineering correlations for turbulent flame speed take the form

\[ \frac{S_T}{S_L} = 1 + C\,\left( \frac{u'}{S_L} \right)^{n} \]

with \( n \) near 1 at low \( u' \) and approaching 0.5 at high \( u' \), reflecting bending of the curve as flamelet interaction and local extinction become important.

6.3 Modeling Approaches

Direct numerical simulation resolves all scales but is limited to low \( Re_t \). Large-eddy simulation resolves the energy-containing motions and models the rest; combustion models include flame surface density, G-equation level-set, thickened flame, and filtered-density-function approaches. RANS is built on Favre-averaged equations with closures such as BML (Bray–Moss–Libby) in the flamelet regime and eddy-dissipation concept for distributed regimes.

7. Diffusion Flames

7.1 Burke–Schumann Analysis

When fuel and oxidizer arrive in separate streams, the flame sits where they meet in stoichiometric proportion. The classical Burke–Schumann treatment assumes infinitely fast chemistry (so fuel and oxidizer cannot coexist) and equal diffusivities. Introducing the conserved Schvab–Zel’dovich variables reduces the coupled fuel, oxidizer, and temperature problem to a single equation for mixture fraction Z. The flame is the surface \( Z = Z_{st} \), and the temperature is piecewise linear in Z with a peak at \( Z_{st} \).

7.2 Flame Sheet, Stretch, and the S-Curve

Real diffusion flames have finite chemistry. Away from extinction, a thin flame-sheet approximation holds, and chemistry corrections are small. As stretch increases (or residence time shrinks), the peak temperature drops below the Burke–Schumann value. Beyond a critical stretch the flame extinguishes and the system jumps to a non-burning, mixed state: the upper branch of the S-shaped response curve of temperature versus Damköhler number collapses. Conversely, a cold mixed state can ignite only when Da is large enough to jump to the upper branch. These two transitions are the ignition and extinction limits of the diffusion flame, and they generalize to partially premixed and stratified flames.

7.3 Counterflow and Jet Flames

The counterflow diffusion flame, with fuel and oxidizer streams impinging axially on a stagnation plane, is the workhorse canonical geometry: it is quasi-one-dimensional along the axis, its strain rate is set by the approach velocity, and it admits well-defined extinction and ignition measurements. Laminar jet diffusion flames (the Burke–Schumann cylindrical duct problem or a free coflow jet) are set by diffusion from the jet into the surrounding oxidizer; flame length scales as

\[ L_f \sim \frac{\dot V}{D} \]

where \( \dot V \) is volumetric flow rate and \( D \) is a diffusivity, independent of jet diameter in the laminar regime. In the turbulent regime flame length is independent of Reynolds number once Re is large, because turbulent mixing replaces molecular diffusion.

7.4 Soot and Smoke Point

Rich diffusion flames produce soot through a sequence of fuel pyrolysis to acetylene, polycyclic aromatic hydrocarbon formation, nucleation into primary soot particles, growth by HACA (hydrogen-abstraction carbon-addition), and coagulation. The smoke point is the fuel flow rate at which a coflow jet flame just begins to emit visible smoke; it is a practical fuel property strongly correlated with aromatic content and C/H ratio.

8. Jet Flames and Flame Stabilization

A lifted jet flame can stabilize at a standoff distance where the local turbulent flame speed matches the local flow speed. The blowoff limit is the velocity beyond which no such balance exists. Burner-stabilized flames use recirculation zones (bluff bodies, swirl, backward-facing steps) to provide a reservoir of hot products that continually re-ignite fresh mixture. Swirl is especially effective: the central recirculation zone behind a strong swirler anchors flames at very high bulk velocities and is the workhorse of modern gas-turbine combustors. The swirl number \( S_n \) is the ratio of angular to axial momentum flux; \( S_n > 0.6 \) is typically required to generate vortex breakdown and the central recirculation zone.

9. Atomization, Droplet, and Spray Combustion

9.1 Atomization

Liquid fuel is burned almost universally as a spray. The atomizer converts a liquid sheet or jet into droplets through aerodynamic shear (airblast), pressure (pressure-swirl/simplex), or pressure–assist. Droplet size distributions are characterized by the Sauter mean diameter \( D_{32} \), the ratio of the third to the second moment of the size distribution, because it couples to evaporation area per unit volume. For a pressure atomizer an empirical correlation reads

\[ D_{32} \propto \sigma^{0.25}\,\mu_L^{0.25}\,\Delta p^{-0.5}\,\rho_L^{0.25} \]

and for airblast atomizers the Rizk–Lefebvre correlation captures the dependence on relative air–liquid velocity, surface tension, and density ratio.

9.2 The \( d^2 \) Law for Droplet Evaporation and Burning

An isolated spherical droplet burning in an oxidizing atmosphere has a steady, quasi-steady gas-phase solution in which the flame sits at a radius \( r_f \) where fuel vapor and oxidizer meet in stoichiometric proportion. Integrating the steady species and energy equations gives

\[ d^2\!\left( t \right) = d_0^2 - K\,t \]

where \( d\!\left( t \right) \) is droplet diameter and K is the burning rate constant,

\[ K = \frac{8\,\rho\,D}{\rho_L}\,\ln\!\left( 1 + B \right) \]

with the Spalding transfer number

\[ B = \frac{Y_{O,\infty}\,Q / s + c_p\,\left( T_\infty - T_s \right)}{L_v} \]

For a hydrocarbon droplet in air at 1 atm, \( K \) is typically 0.7 to 1.2 \( \mathrm{mm}^2\!/\mathrm{s} \). The flame radius satisfies

\[ \frac{r_f}{r_s} = \frac{\ln\!\left( 1 + B \right)}{\ln\!\left( 1 + \left( Y_{O,\infty} / s \right) \right)} \]

and typically sits at 5 to 15 droplet radii for atmospheric pressure combustion.

9.3 Spray Effects and Group Combustion

Real sprays violate the isolated-droplet assumption. The group combustion number

\[ G = 3\,\left( 1 + 0.276\,Re^{1/2} Sc^{1/3} \right) Le\,N^{2/3}\,\frac{r_s}{\ell} \]

(with N droplets in a cluster of characteristic spacing \( \ell \)) classifies the spray as external sheath (G above about 1), internal group, external group, or single-droplet. Dense sprays burn as diffusion flames around the cloud rather than as many individual droplets.

10. Ignition, Quenching, and Extinction

10.1 Thermal Explosion Theory

Semenov’s theory considers a well-stirred mixture in a vessel at wall temperature \( T_w \), with heat release \( Q\,V\,\omega\!\left( T \right) \) and heat loss \( h\,A\,\left( T - T_w \right) \). Steady solutions are intersections of the exponential generation curve and the linear loss line. If they intersect, the lower intersection is a stable slow-reaction state; if they do not (because \( T_w \) is high enough), the system runs away. The critical condition

\[ \frac{E_a\,Q\,V\,\omega\!\left( T_w \right)}{R T_w^2\,h\,A} = e \]

(with the natural constant e on the right) defines the Semenov criterion. Frank-Kamenetskii’s extension treats spatially distributed heat loss with conduction and gives a criterion on the dimensionless group

\[ \delta_{FK} = \frac{E_a\,Q\,\rho\,A\,r^2\,\exp\!\left( -E_a / R T_w \right)}{R T_w^2\,\lambda} \]

exceeding a critical value (2 for a slab, 3.32 for a cylinder, 3.32 for a sphere with appropriate geometry factors). Both theories predict delayed ignition: once threshold is crossed, the induction period is set by the small pre-exponential reaction rate before self-heating takes over.

10.2 Spark Ignition

A spark deposits energy in a small kernel of mixture, heating it to flame temperatures. Whether the kernel grows into a self-propagating flame depends on whether its heat losses (to the wall and to the unburned gas via conduction and radiation) are outpaced by heat release. Lewis and von Elbe’s quenching distance \( \delta_q \) is the gap of parallel plates that prevents flame propagation; it is closely related to flame thickness, with \( \delta_q \approx 2\,\delta_T \) for many mixtures. The MIE scaling of Section 3.2 follows from equating spark energy to the heat capacity of a kernel of volume \( \delta_q^3 \) at flame temperature.

10.3 Autoignition and Homogeneous Charge Compression

Autoignition in a compressed charge (HCCI, diesel) is governed by the ignition-delay correlation of Section 3.3 integrated over the compression history:

\[ \int_0^{t_{ign}} \frac{dt}{\tau_{ign}\!\left( T\!\left( t \right), p\!\left( t \right) \right)} = 1 \]

This Livengood–Wu integral is the standard engineering tool to predict ignition timing in engines. For fuels with NTC behavior, the integral is dominated by the intermediate-temperature interval where \( \tau_{ign} \) is smallest, leading to two-stage ignition.

11. Detonation

11.1 Rankine–Hugoniot and the CJ Point

A detonation is a supersonic combustion wave: a shock wave tightly coupled to a chemical reaction zone. Mass, momentum, and energy conservation across the wave yield the Rankine–Hugoniot relations. In \(\left( 1/\rho, p \right)\) space the locus of possible downstream states is the Hugoniot curve; the heat release shifts it upward from the inert shock Hugoniot. The line connecting the initial state to a downstream state (the Rayleigh line) must have a slope determined by the wave speed. Two Rayleigh lines are tangent to the exothermic Hugoniot: one above the Chapman–Jouguet point (weak detonation, not physically realized from a shock) and one at the CJ point itself. Between them are strong detonations, physically realized with a piston or boundary condition that supports overdriving.

At the upper CJ point the downstream flow relative to the wave is sonic, Ma = 1, which uniquely fixes the detonation velocity \( D_{CJ} \) for a given mixture. For stoichiometric hydrogen–oxygen, \( D_{CJ} \) is near 2800 m/s; for hydrocarbon–air, 1800 to 2000 m/s.

11.2 ZND Structure

The Zel’dovich–von Neumann–Döring model resolves the detonation internal structure: a leading inert shock compresses and heats the mixture to the von Neumann spike (pressure typically twice CJ pressure), followed by a subsonic reaction zone where chemistry proceeds at the nearly constant post-shock temperature. The reaction zone length is \( \delta_{ZND} \sim u\,\tau_{ign}\!\left( T_{vN}, p_{vN} \right) \) and sets the detonation cell size \( \lambda_{cell} \) through empirical correlations \( \lambda_{cell} \approx A\,\delta_{ZND} \) with A of order 30 to 50.

11.3 Deflagration-to-Detonation Transition and Critical Conditions

A subsonic deflagration can accelerate through obstacle-induced turbulence and flame wrinkling until it triggers detonation via local hot spots or shock focusing. The Shchelkin run-up distance is typically tens of tube diameters. The critical tube diameter for sustained detonation to transmit into open space is \( d_c \approx 13\,\lambda_{cell} \). Cell size, measured on smoked foils, is the single most important characterization of a mixture’s detonability.

12. Combustion Aerodynamics

Practical combustors mix large cold and hot streams, stabilize flames in strong flow, and still keep pressure drop below a few percent. The aerodynamic toolkit includes swirl (Section 8), bluff bodies, sudden expansions, coaxial coflow, and film cooling. In gas turbines the primary zone operates fuel-rich to stabilize the flame, then secondary air is added to complete CO oxidation, and dilution air drops exit temperature below turbine metallurgical limits. Liner film cooling uses a thin layer of cold air along the liner surface to keep metal below 1100 K while flame temperatures exceed 2200 K. Lean-premixed and dry-low-emissions combustors push operation to lean \( \phi \) near 0.5 to limit NOx; stability margins are then narrow and combustion instability (thermoacoustic oscillation) becomes a central design problem.

13. Pollutant Formation

13.1 Thermal, Prompt, and Fuel NO

Nitric oxide forms in flames by three main routes. The Zel’dovich (thermal) mechanism,

\[ \mathrm{O} + \mathrm{N}_2 \rightarrow \mathrm{NO} + \mathrm{N} \]\[ \mathrm{N} + \mathrm{O}_2 \rightarrow \mathrm{NO} + \mathrm{O} \]\[ \mathrm{N} + \mathrm{OH} \rightarrow \mathrm{NO} + \mathrm{H} \]

dominates above 1800 K. The first step has an enormous activation energy because N_2 is very stable; this makes thermal NO extremely sensitive to peak temperature, growing roughly exponentially with \( T_b \). The prompt (Fenimore) mechanism starts with CH + N_2 → HCN + N in the flame front and matters chiefly in rich hydrocarbon flames. Fuel NO forms when the fuel itself contains nitrogen (e.g., coal); it proceeds through HCN and NH intermediates with efficiency weakly dependent on temperature.

13.2 CO Emissions

CO is a product of incomplete combustion. It forms readily but oxidizes slowly through CO + OH → CO_2 + H. In quenched regions, lean-blow-out near-limits, or cold boundary layers, OH is unavailable and CO freezes. CO emissions are therefore an indicator of poor mixing or excessive dilution.

13.3 Soot Formation

Soot forms in rich regions through acetylene and PAH chemistry, as described in Section 7.4. Reducing soot requires either avoiding rich zones (lean-premixed combustion) or ensuring soot-laden gas experiences enough high-temperature, oxygen-rich residence time to burn out. For diesel engines, soot–NOx trade-off is the defining emissions challenge: lower temperature reduces NOx but leaves soot unoxidized; higher temperature burns out soot but makes more NOx.

14. Summary and Looking Ahead

ME 557 builds a physically unified view of combustion from chemistry and thermodynamics up through the canonical flames and their device-scale consequences. The central skills acquired are the ability to compute adiabatic and equilibrium states; to read a detailed mechanism and reduce it; to predict laminar flame speed from a reaction–diffusion balance; to classify turbulent flames and choose appropriate models; to set up mixture fraction descriptions of diffusion flames; to apply the \( d^2 \) law and ignition-delay correlations; and to recognize detonation as a shock-coupled limit. ME 658 and other follow-on courses extend the material to advanced turbulent modeling, chemiluminescence diagnostics, high-pressure chemistry, supercritical and transcritical combustion, and emerging low-carbon fuels (hydrogen, ammonia, e-fuels) whose combustion properties differ in important ways from conventional hydrocarbons.