Basics of Computational
Combustion Modelling
Dr. Gavin Tabor
....
Combustion – p.1
Why study Combustion?
Combustion of hydrocarbons – important fuel source
•
Power generation
•
Aircraft, rocket propulsion
•
Cars etc
•
Marine applications
Combustion – rapid chemical reaction between fuel
and oxidant (air). Usually involves fluid flow.
Understand + model processes → improve power
generation, decrease polutants.
....
Combustion – p.2
Physical basis
Complex area of investigation :
•
Combustion affects fluid flow. Heat release
→ ∆T → change physical properties. Also
buoyancy drives flow
•
Fluid flow affects combustion. Brings reactants
together, removes products. Can determine
rate of combustion, and even extinguish flame.
Overall, combustion ≡ rapid chemical reaction
between fuel and oxidiser.
....
However details vary – treat different regimes
differently.
Combustion – p.3
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
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
Examples – non-premixed
Non-premixed :
•
Diesel engines
•
Aircraft gas turbines
•
Furnaces
•
Candles, fires
....
Combustion – p.6
Flame structure
Premixed combustion
– Flame propagating in laminar flow is characterised
by :
Laminar flame speed
sL ,
thickness
lF
Combustion rate controlled by molecular diffusion
processes (DL – diffusivity)
....
Combustion – p.7
Chemistry
Dimensional analysis gives :
DL
lF =
sL
linking these processes.
Chemical reaction time
lF
tL =
sL
....
Combustion – p.8
Turbulent flame structure
This is more complex. Turbulence characterised by
turbulence intensity
u0
Integral, Kolmogorov scales
lI , η
Turbulent Reynolds number
u0 l I
ReT =
ν
Characteristic turbulent eddy turnover time
....
lI
tT = 0
u
Combustion – p.9
Damköhler number
Compare importance of turbulence and combustion –
Damköhler number
tT
l I sL
Da =
= 0
tL
u lF
•
ratio of eddy turnover time to burning
•
effect of turbulence on chemical processes
Other measures :
lF /η
....
u0 /sL
Measure of local distortion of the flame
Relative strength of turbulence.
Combustion – p.10
Flame structure
In more detail, 3 regions :
Oxidant
Inner
Layer
T
Fuel
Oxidation
Layer
Preheat
Zone
....
Flame Propagation
Combustion – p.11
Flame structure
1. a preheat zone,
2. an inner layer of fuel consumption,
3. the oxidation layer.
Thickness of the inner layer
lδ = l F δ
For stoichoimetric methane at 1 atmosphere,
lF = 0.175 mm
and
δ = 0.1
....
Combustion – p.12
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
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
Mass fraction
Define Mass fraction Yi
– the mass of the species per unit mass
of the mixture.
Transport equation
∂ρYi
+ ∇.ρYi u = ∇.ρDi ∇Yi + ρSi
∂t
Source term represents addition/removal due to
combustion
....
Combustion – p.15
Temperature equation
Also need transport equation for some measure
of the energy – say temperature T .
∂ρT
+ ∇.ρT u = ∇.ρD∇T + ρST
∂t
Source term represents radiation, pressure work,
energy release.
....
Combustion – p.16
Reactive scalars
Often group everything together as
“reactive scalars”
ψi = {Y1 , Y2 , . . . , Yn , T }
with transport equation
∂ρψi
+ ∇.ρψi u = ∇.ρDi ∇ψi + ρSψi
∂t
....
Combustion – p.17
Problems
Problems with this approach :
1. n often quite large!
– detail 100’s of
reactions for combustion process. Invoke
QSSA to reduce to manageable proportions
(1-5).
Elementary reaction mechanisms
2. Turbulence still unaccounted for!!
Turbulence – introduces too many details to
calculate. Use Reynolds averaging to eliminate
details, replace by a model.
....
Combustion – p.18
Moment methods
Turbulence modelling – replace ‘small scale’ detail
of 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) = u
fx + u00x
Here
ρux = 0
....
and thus
ρux
u
fx =
ρ
Combustion – p.19
Favre averaging
Useful for modelling when density varies : NSE
contain terms such as
ρux uy
Using Favre averaging this becomes
00 u00
ρux uy = ρf
ux u
fy + ρug
x y
....
Use this to derive mean flow equations
(Favre-averaged NSE) and equations for k and – a turbulence model for compressible flow.
Combustion – p.20
Moment methods cont.
Favre averaged equation for reactive scalars :
∂ρψei
00 ψ 00 + ρS
f
e = ∇.ρDi ∇ψi − ∇.ρug
+ ∇.ρψei u
ψi
i
∂t
High Re, so molecular diffusivity D can be ignored.
Two terms cause problems :
00 ψ 00
−∇.ρug
i
representing turbulent transport
and
....
ρSf
ψi
the mean chemical source term
Combustion – p.21
Specific models
Different combustion models arrise from different
approaches to these terms – range from cheap and
inaccurate 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 − model for turbulence
•
cheap to compute. Requires extensive tuning.
....
Combustion – p.22
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
Flamelet methods
Main alternative methodology.
High Re, Da ⇒ flame fronts very thin – consider
as 2d sheet which :
....
•
separates burnt and unburnt gas (premixed
combustion)
•
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
Indicator function
Premixed combustion : progress variable
T − Tu
c=
Tb − T u
or
YP
c=
YP,b
0 < c < 1 : 1 represents burned gas, 0 unburned.
Some intermediate value represents flame centre.
....
Combustion – p.25
1-d representation
Plotting c across the flame :
1
c
Unburned
Burned
0
c = 0.5 = flame centre
....
Combustion – p.26
Flame wrinking
Flame will be wrinkled by turbulence on scales too
small to simulate
....
Combustion – p.27
Averaging
Average equation
∂ρe
c
00 c00 + ρS
ec
+ ∇.ρe
cu = −∇.ρug
∂t
Need to model :
00 c00 turbulent transport term
ug
Sec reaction term
– but now have a manageable set of equations.
....
Combustion – p.28
Other models
Other versions
• flame surface density Σ
• G -equation
•
Non-premixed combustion
– use mixture fraction Z .
....
Combustion – p.29
Counter-gradient transport
00 c00 often uses gradient
Final note : modelling of ug
transport assumption
00 c00 = −D ∇e
ug
t c
However this is frequently wrong – counter-gradient
transport.
....
Combustion – p.30