AME 436 Energy and Propulsion

1
AME 436Energy and Propulsion

• Lecture 4
• Basics of combustion

2
Outline

• Why do we need to study combustion?
• Types of flames - premixed and nonpremixed
• Basics of chemical reaction rates
• Law Of Mass Action (LOMA)
• Arrhenius form of temperature dependence
• Premixed flames
• Deflagrations - burning velocity, flame
  thickness, temperature effect
• Turbulence effects
• Homogeneous reaction
• Nonpremixed flames
• General characteristics
• Droplets
• Gas-jet
• Turbulence effects

3
Why do we need to study combustion?

• Chemical thermodynamics only tells us the end
  states - what happens if we wait forever and a
  day for chemical reaction to occur
• We also need to know how fast reactions occur
• How fast depends on both the inherent rates of
  reaction and the rates of heat and mass transport
  to the reaction zone(s)
• Chemical reactions heat mass transport
  combustion
• Some reactions occur too slowly to be of any
  consequence, e.g.
• 2 NO ? N2 O2
• has an adiabatic flame temperature of 2869K (no
  dissociation) or 2650K (with dissociation, mostly
  NO O) but no one has ever made a flame with NO
  because reaction rates are too slow!
• What do we do with this information in the
  context of engines?
• Determine rates of flame propagation and heat
  generation
• Determine conditions for knock in
  premixed-charge engines
• Determine rates of pollutant formation and
  destruction

4
Types of flames

• Premixed - reactants are intimately mixed on the
  molecular scale before combustion is initiated
  several flavors
• Deflagration
• Detonation
• Homogeneous reaction
• Nonpremixed - reactants mix only at the time of
  combustion - have to mix first then burn several
  flavors
• Gas jet (Bic lighter)
• Liquid fuel droplet
• Liquid fuel jet (e.g. Kuwait oil fire, candle)
• Solid (e.g. coal particle, wood)

5
Premixed flames - deflagration

• Propagating subsonic front sustained by
  conduction of heat from the hot (burned) gases to
  the cold (unburned) gases which raises the
  temperature enough that chemical reaction can
  occur
• Since chemical reaction rates are very sensitive
  to temperature, most of the reaction is
  concentrated in a thin zone near the
  high-temperature side
• May be laminar or turbulent

Turbulent premixed flame experiment in a
fan-stirred chamber (http//www.mech-eng.leeds.ac.
uk/res-group/combustion/activities/Bomb.htm)
6
Premixed flames - detonation

• Supersonic propagating front sustained by heating
  of gas by a shock wave
• After shock front, need time (thus distance
  time x velocity) before reaction starts to occur
  (induction zone)
• After induction zone, chemical reaction heat
  release occur
• Pressure temperature behavior coupled strongly
  with supersonic/subsonic gasdynamics
• Ideally only M3 1 Chapman-Jouget detonation
  is stable
• (M Mach number Vc V velocity,
• c sound speed (?RT)1/2 for ideal gas)

7
Premixed flames - homogeneous reaction

• Model for knock in premixed-charge engines
• Fixed mass (control mass) with uniform (in space)
  T, P and composition
• No propagation in space but propagation in time
• In laboratory, we might heat the chamber to a
  certain T and see how long it took to react in
  engine, compression of mixture (increases P T,
  thus reaction rate) will initiate reaction

8
Non-premixed or diffusion flames

• Only subsonic
• Generally assume mixed is burned - mixing
  slower than chemical reaction

9
? (??)1/2
10
Law of Mass Action (LoMA)

• First we need to describe rates of chemical
  reaction
• For a chemical reaction of the form
• ?AA ?BB ? ?CC ?DD
• e.g. 1 H2 1 I2 ? 2 HI
• A H2, ?A 1, B I2, ?B 1, C HI, ?C 2, D
  nothing, ?D 0
• the Law of Mass Action (LoMA) states that the
  rate of reaction
• i concentration of molecule i (usually
  moles per liter)
• kf forward reaction rate constant
• How to calculate i ?
• According to ideal gas law, the total moles of
  gas per unit volume (all molecules, not just type
  i) P/?T
• Then i (Total moles / volume)(moles i /
  total moles), thus
• i (P/?T)Xi (Xi mole fraction of i
  (see lecture 2))
• Minus sign on dA/dt and dB/dt since A B are
  being depleted
• Basically LoMA states that the rate of reaction
  is proportional to the number of collisions
  between the reactant molecules, which in turn is
  proportional to the concentration of each
  reactant

11
Comments on LoMA

• The reaction rate constant kf is usually of the
  Arrhenius form
• Z pre-exponential factor, a another
  (nameless) constant, E activation energy
  (cal/mole) working backwards, units of Z must be
  (moles per liter)1-?A-vB/(K-asec)
• With 3 parameters (Z, n, E) any curve can be fit!
• The exponential term causes extreme sensitivity
  to T for E/? gtgt T!

12
Comments on LoMA

• Boltzman (1800s) showed that the fraction of
  molecules in a gas with translational kinetic
  energy greater than some value E is proportional
  to exp(-E/?T), thus E represents the energy
  barrier that must be overcome for reaction to
  occur
• Note that E is not the same thing as enthalpy of
  reaction ?hf (or heating value QR) and in general
  the two have no relation to each other - E
  affects reaction rates whereas ?hf QR affect
  end states (e.g. Tad) - of course ?hf QR affect
  reaction rates indirectly by affecting T
• Diary of a collision

13
Comments on LoMA

• The full reaction rate expression is then
• H2 I2 ? 2HI is one of few examples where the
  actual conversion of reactants to products occurs
  in a single step most fuels of interest go
  through many intermediates during oxidation even
  for the simplest hydrocarbon (CH4) the standard
  mechanism (http//www.me.berkeley.edu/gri_mech/)
  includes 53 species and 325 individual reactions!
• The only likely reactions in gases, where the
  molecules are far apart compared to their size,
  are 1-body, 2-body or 3-body reactions, i.e. A ?
  products, A B ? products or A B C ?
  products
• In liquid or solid phases, the close proximity of
  molecules makes n-body reactions plausible

14
Comments on LoMA

• Recall that the forward reaction rate is
• Similarly, the rate of the reverse reaction can
  be written as
• kb backward reaction rate constant
• At equilibrium, the forward and reverse rates
  must be equal, thus

15
Deflagrations - burning velocity

• Since the burning velocity (SL) ltlt sound speed,
  the pressure across the front is almost constant
• How fast will the flame propagate? Simplest
  estimate based on the hypothesis that
• Rate of heat conducted from hot gas to cold gas
  (i)
• Rate at which enthalpy is conducted through flame
  front (ii)
• Rate at which heat is produced by chemical
  reaction (iii)

16
Deflagrations - burning velocity

• Estimate of i
• Conduction heat transfer rate -kA(?T/?)
• k gas thermal conductivity, A cross-sectional
  area of flame
• ?T temperature rise across front Tproducts -
  Treactants Tad - T8
• ? thickness of front (unknown at this point)
• Estimate of ii
• Enthalpy flux through front (mass flux) x Cp x
  ?T
• Mass flux ?VA (? density of reactants ?8, V
  velocity SL)
• Enthalpy flux ?8CpSLA?T
• Estimate of iii
• Heat generated by reaction QR x (dfuel/dt) x
  Mfuel x Volume
• Volume A?
• QR CP?T/f

17
Burning velocity, flame thickness

• Combine (i) and (ii)
• ? k/?CpSL ?/SL (? flame thickness)
• ? k/?Cp thermal diffusivity (units
  length2/time)
• For air at 300K 1 atm, ? 0.2 cm2/s
• For gases ? ? D (? kinematic viscosity D
  mass diffusivity)
• For gases ? P-1T1.7 since k P0T.7, ? P1T-1,
  Cp P0T0
• For typical stoichiometric hydrocarbon-air flame,
  SL 40 cm/s, thus ? ?/SL 0.005 cm (!)
  (Actually when properties are temperature-averaged
  , ? 4?/SL 0.02 cm - still small!)
• Combine (ii) and (iii)
• SL ??1/2
• ? overall reaction rate (dfuel/dt)/fuel8
  (units 1/s)
• With SL 40 cm/s, ? 0.2 cm2/s, ? 1600 s-1
• 1/? characteristic reaction time 625
  microseconds
• Heat release rate per unit volume (enthalpy
  flux) / (volume) (?CpSLA?T)/(A?)
  (?CpSL/k)(k?T)/? (k?T)/?2
• (0.07 W/mK)(1900K)/(0.0002 m)2 3 x 109 W/m3
  !!!
• Moral flames are thin, fast and generate a lot
  of heat!

18
Deflagrations - burning velocity

• More rigorous analysis (Zeldovich, 1940)
• Same functional form as simple estimate (SL
  ??1/2, where ? is an overall reaction rate)
  with some additional constants
• How does SL vary with pressure?
• Define order of reaction (n) ?A ?B since
• Thus SL ??1/2 P-1Pn-11/2 P(n-2)/2
• For typical n 2, SL independent of pressure
• For real hydrocarbons, working backwards from
  experimental results, we find typically SL
  P-0.1, thus n 1.8

19
Deflagrations - temperature effect

• Since Zeldovich number (b) gtgt 1
• For typical hydrocarbon-air flames, E 40
  kcal/mole
• ? 1.987 cal/mole, Tad 2200K
• ? b 10, at T close to Tad, ? T10 !!!
• ? Thin reaction zone concentrated near
  highest temperature
• ? In Zeldovich (or any) estimate of SL, overall
  reaction rate ? must be evaluated at Tad, not T8
  or any other temperature
• How can we estimate E? If reaction rate depends
  more on E than concentrations , SL ??1/2
  exp(-E/?Tad)1/2
• exp(?E/2?Tad) - Plot of ln(SL) vs. 1/Tad has
  slope of -E/2?
• If b isnt large, then ?(T8) ?(Tad) and
  reaction occurs even in the cold gases, so no
  control over flame is possible!
• Since SL ?1/2, SL (Tadb)1/2 Tad5 typically!

20
Deflagrations - summary

• These relations show the effect of Tad (depends
  on fuel stoichiometry), ? (depends on diluent
  gas (usually N2) P), ? (depends on fuel, T, P)
  and pressure (engine condition) on laminar
  burning rates
• Re-emphasize these estimates are based on an
  overall reaction rate real flames have 1000s of
  individual reactions between 100s of species -
  but we can work backwards from experiments or
  detailed calculations to get these estimates for
  the overall reaction rate parameters

21
Deflagrations
Schematic of flame temperatures and laminar
burning velocities
Real data on SL (Vagelopoulos Egolfopoulos,
1998)
22
Turbulent flames - motivation

• Almost all flames used in practical combustion
  devices are turbulent because turbulent mixing
  increases burning rates, allowing more
  power/volume
• Examples
• Premixed turbulent flames
• Gasoline-type (spark ignition, premixed-charge)
  internal combustion engines
• Stationary gas turbines (used for power
  generation, not propulsion)
• Nonpremixed flames
• Diesel-type (compression ignition,
  nonpremixed-charge) internal combustion engines
• Gas turbines
• Most industrial boilers and furnaces

23
Basics of turbulence

• Good reference Tennekes A First Course in
  Turbulence
• Job 1 need a measure of the strength of
  turbulence
• Define turbulence intensity (u) as rms
  fluctuation of instantaneous velocity u(t) about
  mean velocity ( )
• Kinetic energy of turbulence massu2/2 KE per
  unit mass (total in all 3 coordinate directions
  x, y, z) 3u2/2
• Note that for a typical u/SL 5, with SL 40
  cm/s, u 2 m/s, thus KE 6 m2/s2 6
  (kg-m2/s2)/kg 6 J/kg
• Is this KE large or small? Typical hydrocarbon
• f 0.062, QR 4.3 x 107 J/kg, thus for
  fuel-air MIXTURE, energy/mass 0.062 4.3 x
  107 J/kg 2.7 x 106 J/kg
• i.e. 444,000 times larger than KE of turbulence
• Moral it pays to be turbulent (but not so
  turbulent that flame quenching occurs as
  discussed later)

24
Basics of turbulence

• Job 2 need a measure of the length scale of
  turbulence
• Define integral length scale (LI) as
• Here the overbars denote spatial (not temporal)
  averages
• LI is a measure of size of largest eddies, i.e.
  the largest scale over which velocities are
  correlated
• Typically related to size of system (tube or jet
  diameter, grid spacing, )
• A(r) is the autocorrelation function at some time
  t
• Note A(0) 1 (fluctuations around the mean are
  perfectly correlated at a point)
• Note A(8) 0 (fluctuations around the mean are
  perfectly uncorrelated if the two points are very
  distant)
• For truly random process, A(r) is an
  exponentially decaying function
• A(r) exp(-r/LI)

25
Basics of turbulence

• In real experiments, generally know u(t) not u(x)
  - can define time autocorrelation function A(x,?)
  and integral time scale ?I at a point x
• Here the overbars denote temporal (not spatial)
  averages
• With suitable assumptions LI (8/p)1/2u?I
• Define integral scale Reynolds number ReL ?
  uLI/?
• (? kinematic viscosity)
• Note generally ReL ? Reflow Vd/? typically u
  0.1V, LI 0.5d, thus ReL 0.05 Reflow (e.g.
  in pipe-flow turbulence, grid turbulence, flow
  behind a cylinder, etc.)

26
Characteristics of turbulent flames

• Most important property turbulent flame speed
  (ST)
• Most models based on physical models introduced
  by Damköhler (1940)
• Behavior depends on Karlovitz number - ratio of
  turbulent strain rate to chemical rate for
  standard Kolmogorov turbulence model
• Low Ka Huygens propagation, thin fronts that
  are wrinkled by turbulence but internal structure
  is unchanged
• High Ka Distributed reaction zones, broad
  fronts
• Note at low u/SL, ST/SL increases rapidly with
  increasing u/SL, but at high enough u/SL there
  is almost no increase in ST/SL and in fact flame
  quenching occurs at sufficiently high u/SL (thus
  high Ka (u/SL)2)

27
Characteristics of turbulent flames
28
Bradley et al. (1992)

• Compilation of data from many sources

ST/SL
u/SL
29
Turbulent burning velocity

• Experimental results shown in Bradley et al.
  (1992) based on smoothed data from many sources,
  e.g. fan-stirred bomb

30
Bradley et al. (1992)

• and we are talking MAJOR smoothing!

31
Turbulent premixed flame modeling

• Thin-flame behavior observed in most practical
  combustors
• Damköhler (1940) in Huygens propagation regime,
  flame front is wrinkled by turbulence but
  internal structure and SL are unchanged
• Propagation rate ST due only to area increase via
  wrinkling ST/SL AT/AL
• Many models, still much controversy about how to
  model, but most show ST/SL u/SL

32
Turbulent premixed combustion

• Models of premixed turbulent combustion dont
  agree with experiments nor each other!
• All models are trying to predict the same thing

33
Turbulent premixed flame modeling

• Low u/SL weakly wrinkled flames
• ST/SL 1 (u/SL)2 (Clavin Williams, 1979) -
  standard for many years
• Actually Kerstein and Ashurst (1994) showed this
  is valid only for periodic flows - for random
  flows ST/SL - 1 (u/SL)4/3
• Higher u/SL strongly wrinkled flames
• Schelkin (1947) - AT/AL estimated from ratio of
  cone surface area to base area height of cone
  u/SL result
• Yahkot (1988)
• Other models based on fractals,
  probability-density functions, etc. most predict
  ST/SL u/SL at high u/SL
• For the purposes of this class well usually
  assume ST/SL u/SL with the possibility of
  bending or quenching at sufficiently high Ka
  (u/SL)2

34
Turbulence in engines

• How to get high turbulence in engines?
• Geometry of intake valves, ports, etc. - cause
  gas to swirl as it enters combustion chamber
• Cup-shaped piston head (squish)
• Obstacles in flow

35
Homogenous reaction

• Given a homogenous system (T, P, same
  everywhere at any instant in time, but may change
  over time), how long will it take for the mixture
  to react (explode?)
• Model for knocking in premixed-charge piston
  engines
• As reaction starts, heat is released, temperature
  increases, overall reaction rate ? increases,
  heat is released faster, T rises faster, ?
  increases faster, ltBOOMgt
• Simple analysis - assumptions
• Single-step reaction ?AA ?BB ? ?CC ?DD
• Excess of B (example lean mixture with A
  fuel, B oxygen)
• ?A ?B 1
• Adiabatic, constant-volume, ideal gas, constant
  Cv
• Constant mass

36
Homogenous reaction

• Energy equation - if all fuel consumed
• (Yf fuel mass fraction)
• So at any instant in time
• where Yf(t) is the instantaneous fuel mass
  fraction (at t 0, no fuel consumed, T initial
  temperature T8 at t 8, Yf 0, all fuel
  consumed, T Tad) then from page 16
• (this simply says that there is a linear
  relationship between the amount of fuel consumed
  and the temperature rise)
• Since we assumed ?A ?B 1, where A fuel, B
  oxygen

37
Homogenous reaction

• Reaction rate equation (assume n in RR expression
  0)
• Combine Eqs. 1, 2, 3, non-dimensionalize
• Notes on this result
• ? is the equivalence ratio for our special case
  ?A ?B 1 only valid for lean mixtures since
  we assumed surplus of A fuel
• Get pressure from P(t) ?8RT(t)

38
Homogenous reaction

• This equation looks scary but its just a 1st
  order nonlinear ordinary differential equation -
  can integrate to find ?(?) (amount of reaction
  product formed as a function of time) for various
  values of the parameters ? (stoichiometry), ?
  (activation energy, initial temperature T8), H
  (heat release)
• Initial condition is ? 1 at ? 0
• What do we expect?
• Since reaction rate is slowest at low T, reaction
  starts slowly then accelerates
• Induction time (e.g. time to reach 90
  completion of reaction, ? 0.1) should depend
  mostly on initial temperature T8, not final
  temperature Tad since most of the time needed to
  react is before self-acceleration occurs
• This is very different from propagating flames
  where SL depends mostly on Tad not T8 - why?
  Because in the flame case there was a source of
  high temperature (the burned gases) to raise the
  gas up to near Tad before reaction started in
  the homogenous case there is no such source
• This means that the factors that affect flame
  propagation and knock are very different

39
Homogenous reaction

• Double-click chart to edit or change parameters
• Case shown ? 0.7, ? 10, H 6
• Note profile and time to ignite depend strongly
  on ?, much less on ? and H

40
Homogenous reaction

• In case of real chemistry, besides the thermal
  acceleration mechanism there is also a chemical
  or chain branching acceleration mechanism, e.g.
  for H2-O2
• H O2 ? OH O
• H2 OH ? H H2O
• O H2 ? OH H etc.
• where 1 radical (H, OH, O) leads to 2, then 4,
  then 8, radicals
• In the case above, the net reaction would be
• 2 H2 O2 ? H OH H2O
• which shows the increase in the radical pool
• This chain branching mechanism leads to even
  faster runaway than thermal runaway since 2x gt
  e-a/x for sufficiently large x
• What if no H to start with?
• H2 O2 ? HO2 H (mostly this - ?h 55
  kcal/mole )
• H2 M ? H H M (slower since ?h 104
  kcal/mole - too big)
• O2 M ? O O M (?h 119 kcal/mole - even
  worse)
• (M any molecule)

41
Non-premixed or diffusion flames

• Simplest approach to determining properties
  mixed is burned - chemical reaction rates
  faster than mixing rates
• No inherent propagation rate (unlike premixed
  flames where SL ??1/2)
• No inherent thickness ? (unlike premixed flames
  where thickness ?/SL) - in nonpremixed flames,
  determined by equating convection time scale
  ?/V ? to diffusion time scale ?2/? ? ?
  (??)1/2 where ? is a characteristic flow time
  scale (e.g. d/V for a jet, where d diameter, V
  velocity, or LI/u for turbulent flow, etc.)
• Burning must occur near stoichiometric contour
  where reactant fluxes are in stoichiometric
  proportions (otherwise surplus of one reactant)
• Burning must occur near highest T since ?
  exp(-E/RT) is very sensitive to temperature (like
  premixed flames)

42
Non-premixed or diffusion flames

• Well look at two examples of non-premixed flames
  which represent opposite extremes of what might
  happen in a Diesel engine
• Droplet combustion - vaporization of droplets is
  slow, so droplets burn as individuals
• Gas-jet flame - vaporization of droplets is so
  fast, there is effectively a jet of fuel vapor
  rather than individual droplets
• Reality is in between, but in Diesels usually
  closer to the gas jet with extras

Flynn, P.F, R.P. Durrett, G.L. Hunter, A.O. zur
Loye, O.C. Akinyemi, J.E. Dec, C.K. Westbrook,
SAE Paper No. 1999-01-0509.
43
Droplet combustion

• Heat from flame is conducted to fuel surface,
  vaporizes fuel, fuel convects/diffuses to flame
  front, O2 diffuses to flame front from outside,
  burning occurs at stoichiometric location
• As fuel burns, droplet diameter d(t) decreases
  until d 0 or droplet may extinguish before
  reaching d 0
• Experiments typically show d(0)2 - d(t)2 Kt
• Model for droplets in Diesel engine combustion

44
Droplet combustion

• How fast does droplet burn? Spherically-symmetric
  model (Godsave, Spalding 1953), assuming mixed
  is burned, affectionately called the
  dee-squared law
• d(0) droplet radius at time 0
• d(t) droplet radius at some later time t
• K droplet burning rate constant (m2/s)
• k gas thermal conductivity (W/mK)
• ?l droplet density (kg/m3)
• CP gas specific heat at constant pressure
  (J/kgK)
• QR fuel heating value per unit mass (J/kg)
• f stoichiometric fuel mass fraction
• T8 ambient air temperature (K)
• Td droplet vaporization temperature (K)
• Lv latent heat of vaporization of fuel (J/kg)
• B (Transfer number) ratio of heat generation
  by chemical reaction to heat needed to vaporize
  fuel typical values methanol 3, most
  hydrocarbons 8 - 10
• Diameter of flame surrounding droplet (dflame)

45
Droplet combustion

• The d2-law assumes no buoyant or forced
  convection, but in engines there is likely to be
  a lot of flow one relation for the effect of
  flow on burning rate is
• Red Droplet Reynolds number Ud(t)/?
• Nu Nusselt number based on droplet diameter
• U droplet velocity relative to gas ?/?
• Pr Prandtl number ?/?
• ? kinematic viscosity
• ? thermal diffusivity kg/?gCp,g
• Note that this result reduces to the previous one
  for U 0 (thus Re 0)

46
Droplet combustion

• Note all the heat release (QR), heat of
  vaporization (Lv), etc. is tied up in B which
  appears only inside a ln( ), thus changing these
  properties hardly affects burning rate at all
• Why? The more you vaporize fuel, the more
  rapidly the fuel vapor blows out, thus the harder
  it is for heat to be conducted to the fuel
  surface

Marchese et al. (1999), space experiments,
heptane in O2-He
47
Nonpremixed-gas flames - laminar gas-jet flames

• Flame height (Lf) determined by equating
  diffusion time (dj2/D, dj jet diameter, D
  oxygen diffusivity) to convection time (Lf/V) (V
  jet exit velocity)
• dj2/D Lf/V ? Lf Vdj2/D or Lf/dj Vdj/D
• Gases D ? ? Lf/dj Vdj/? Red
• Consistent with more rigorous models

48
Nonpremixed turbulent jet flames

• Turbulent (Hottel and Hawthorne, 1949)
• For turbulent flows D is not constant but rather
  D uLI
• u V LI dj ? Lf dj (independent of Re)
• High V ? high u ? Ka large - flame lifts off
  near base
• Still higher - more of flame lifted
• When lift-off height flame height, flame blows
  off (completely extinguished)

Lifted flame (green fuel blue flame)
49
Summary - combustion

• Combustion is the combination of chemical
  reaction with convective and diffusive transport
  of thermal energy and chemical species
• The most important distinction between flames is
  premixed vs. non-premixed, i.e. whether the
  reactants are mixed before combustion
• Chemical reactions relevant to combustion are
  generally VERY complicated but can often be
  approximated (roughly) by a one step overall
  reaction
• Chemical reactions relevant to combustion
  generally have high activation energy (more
  precisely, high Zeldovich number b) and thus are
  more sensitive to temperature than any other
  property
• Premixed flames
• Deflagrations - subsonic - burning velocity SL
  (??)1/2 (? reaction rate at Tad)
• Detonations - supersonic wave
• Homogenous reaction - time of reaction depends
  mainly on T8 not Tad
• Nonpremixed flames
• Mixed is burned
• Turbulence increases the rates of combustion by
  increasing surface area (premixed) or mixing
  rates (nonpremixed)