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Basics of ComputationalCombustion Modelling
Dr. Gavin Tabor
Combustion – p.1
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Why study Combustion?
Combustion of hydrocarbons – important fuel source• Power generation• Aircraft, rocket propulsion• Cars etc• Marine applications
Combustion – rapid chemical reaction between fueland oxidant (air). Usually involves fluid flow.
Understand + model processes → improve powergeneration, decrease polutants.
Combustion – p.2
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Physical basis
Complex area of investigation :• Combustion affects fluid flow. Heat release→ ∆T → change physical properties. Alsobuoyancy drives flow
• Fluid flow affects combustion. Brings reactantstogether, removes products. Can determinerate of combustion, and even extinguish flame.
Overall, combustion ≡ rapid chemical reactionbetween fuel and oxidiser.
However details vary – treat different regimesdifferently.
Combustion – p.3
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Combusion regimes
• Non-premixed combustion• Regions of fuel + oxidiser separate,• Combustion occurs at interface
• Premixed combustion• Fuel and oxidiser mixed at molecular level• Regions of burned and unburned gas,
separated by (thin) flame• Partially premixed combustion
• somewhere between
Combustion – p.4
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Some examples
Premixed :• Spark ignition (IC) petrol engines• Lean burn gas turbines• Household burners• Bunsen burner (blue flame regime)
Partially premixed :• Direct injection (DI) petrol engines• Aircraft gas turbines
Combustion – p.5
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Examples – non-premixed
Non-premixed :• Diesel engines• Aircraft gas turbines• Furnaces• Candles, fires
Combustion – p.6
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Flame structure
Premixed combustion
– Flame propagating in laminar flow is characterisedby :
Laminar flame speed sL,
thickness lF
Combustion rate controlled by molecular diffusionprocesses (DL – diffusivity)
Combustion – p.7
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Chemistry
Dimensional analysis gives :
lF =DL
sL
linking these processes.
Chemical reaction time
tL =lFsL
Combustion – p.8
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Turbulent flame structure
This is more complex. Turbulence characterised by
turbulence intensity u′
Integral, Kolmogorov scales lI , η
Turbulent Reynolds number ReT =u′lIν
Characteristic turbulent eddy turnover time
tT =lIu′
Combustion – p.9
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Damköhler number
Compare importance of turbulence and combustion –Damköhler number
Da =tTtL
=lIsLu′lF
• ratio of eddy turnover time to burning• effect of turbulence on chemical processes
Other measures :
lF /η Measure of local distortion of the flame
u′/sL Relative strength of turbulence.
Combustion – p.10
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Flame structure
In more detail, 3 regions :
Oxidant
Fuel
T
PreheatZone
InnerLayer
OxidationLayer
Flame Propagation
Combustion – p.11
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Flame structure
1. a preheat zone,
2. an inner layer of fuel consumption,
3. the oxidation layer.
Thickness of the inner layer
lδ = lF δ
For stoichoimetric methane at 1 atmosphere,
lF = 0.175 mm and δ = 0.1
Combustion – p.12
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Non-premixed combustion
• No obvious characteristic velocity scale• Define a characteristic diffusion thickness lD• Flame still divided into fuel consumption and
oxidation layers• Oxidation layer
lε = εlD
of more importance.
Combustion – p.13
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Modelling – direct approach
We can approach the problem in a simple way :• Combustion ≡ chemical reaction process
combining reactants to form products• Write balance equations for species• Solve together with continuity and momentum
equations
Combustion – p.14
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Mass fraction
Define Mass fraction Yi
– the mass of the species per unit massof the mixture.
Transport equation
∂ρYi∂t
+ ∇.ρYiu = ∇.ρDi∇Yi + ρSi
Source term represents addition/removal due tocombustion
Combustion – p.15
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Temperature equation
Also need transport equation for some measureof the energy – say temperature T .
∂ρT
∂t+ ∇.ρTu = ∇.ρD∇T + ρST
Source term represents radiation, pressure work,energy release.
Combustion – p.16
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Reactive scalars
Often group everything together as“reactive scalars”
ψi = {Y1, Y2, . . . , Yn, T}
with transport equation
∂ρψi∂t
+ ∇.ρψiu = ∇.ρDi∇ψi + ρSψi
Combustion – p.17
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Problems
Problems with this approach :
1. n often quite large!Elementary reaction mechanisms – detail 100’s ofreactions for combustion process. InvokeQSSA to reduce to manageable proportions(1-5).
2. Turbulence still unaccounted for!!Turbulence – introduces too many details tocalculate. Use Reynolds averaging to eliminatedetails, replace by a model.
Combustion – p.18
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Moment methods
Turbulence modelling – replace ‘small scale’ detailof turbulence with (cheaper) turbulence model.
Similar process used in combustion modelling –average to remove details, then substitute a model.
Density of fluid variable ⇒ use Favre averaging.
ux(x, t) = ux + u′′x
Here
ρux = 0 and thus ux =ρuxρ
Combustion – p.19
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Favre averaging
Useful for modelling when density varies : NSEcontain terms such as
ρuxuy
Using Favre averaging this becomes
ρuxuy = ρuxuy + ρu′′xu′′
y
Use this to derive mean flow equations(Favre-averaged NSE) and equations for k and ε– a turbulence model for compressible flow.
Combustion – p.20
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Moment methods cont.
Favre averaged equation for reactive scalars :
∂ρψi∂t
+ ∇.ρψiu = ∇.ρDi∇ψi −∇.ρu′′ψ′′
i + ρSψi
High Re, so molecular diffusivity D can be ignored.Two terms cause problems :
−∇.ρu′′ψ′′
i representing turbulent transport
and
ρSψithe mean chemical source term
Combustion – p.21
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Specific models
Different combustion models arrise from differentapproaches to these terms – range from cheap andinaccurate to precise and expensive.
E.g. Eddy Breakup Model – Spalding (1971)• assumes turbulent mixing determines chemical
reaction rate• gives simple model for chemical source term• combine with k − ε model for turbulence• cheap to compute. Requires extensive tuning.
Combustion – p.22
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PDF Transport
Probability Distribution Function methods• turbulence can be described/modelled in
terms of correlation functions• turbulent processes – combustion – can also be
described/modelled in terms of correlations• joint PDF of velocity and reactive scalars
P (u, ψ;x, t)
• Potentially more accurate. Also very expensive.
Combustion – p.23
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Flamelet methods
Main alternative methodology.
High Re,Da ⇒ flame fronts very thin – consideras 2d sheet which :
• separates burnt and unburnt gas (premixedcombustion)
• separates fuel and air (non-premixed)• is distorted by mean flow and by turbulence• propagates by burning.
Location of flame sheet specified by indicator function.
Combustion – p.24
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Indicator function
Premixed combustion : progress variable
c =T − TuTb − Tu
or c =YPYP,b
0 < c < 1 : 1 represents burned gas, 0 unburned.
Some intermediate value represents flame centre.
Combustion – p.25
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1-d representation
Plotting c across the flame :
c
c = 0.5 = flame centre
0
1
BurnedUnburned
Combustion – p.26
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Flame wrinking
Flame will be wrinkled by turbulence on scales toosmall to simulate
Combustion – p.27
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Averaging
Average equation
∂ρc
∂t+ ∇.ρcu = −∇.ρu′′c′′ + ρSc
Need to model :
u′′c′′turbulent transport term
Sc reaction term
– but now have a manageable set of equations.
Combustion – p.28
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Other models
Other versions• flame surface density Σ
• G-equation• Non-premixed combustion
– use mixture fraction Z.
Combustion – p.29
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Counter-gradient transport
Final note : modelling of u′′c′′ often uses gradienttransport assumption
u′′c′′ = −Dt∇c
However this is frequently wrong – counter-gradienttransport.
Combustion – p.30
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