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Detailed Flame ModelIIT Kanpur

• Detailed chemical kinetic mechanism(elementary reactions)

• Not assuming temperature profile

• Solve using computer

∆∆

• Mass conservation: = 0 =

• 1-D

• Steady state

• Neglect KE and PE terms

• Pressure is constant

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Detailed Flame Model (Formulation)IIT Kanpur

• Species conservation ∆∆( ) ( ) ∆ =

, = constant (mass conservation) = + ,

bulk flow diffusion

+ , = For: 1, 2, …, N species

, = , bulkvelocity

speciesvelocity

∑diffusionalvelocity −velocity of individual species

relative to bulk velocity

for constant diffusivity−

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Detailed Flame Model (Formulation)IIT Kanpur

• Energy conservation ∆ 0, no heat loss

Tx Tx+x

[ ] [ ] ∆

0, no shaft work

ℎ + + ℎ + + ∆0, KE & PEneglected

0, KE & PEneglected

constant

heatflux

( ∆ − ) + (ℎ ∆ − ℎ ) = 0 + = 0 both conduction and additional enthalpy flux from diffusing species = − + ∑ , ℎheatflux

conductioncontribution

species diffusioncontribution , = − = − + ∑ − ℎ = − + ∑ ℎ − ∑ ℎ constant

ℎ

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Detailed Flame Model (Formulation)IIT Kanpur = − + ∑ ℎ − ℎ

+ = 0substitute in energy equation:

− + ∑ ℎ − + = 0− + ∑ + ∑ℎ = 0 (species conservation)− + ∑ = −∑ℎ ℎ = ℎ , + ∫ ,= ∫ , = ,− + ∑ , = −∑ℎ

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Detailed Flame Model (Formulation)IIT Kanpur − + ∑ , = −∑ℎ + ,− + ∑ + , , = −∑ℎ

constant (mass conservation)

+ − + ∑ , , = −∑ℎ − + ∑ , + ∑ , , = −∑ℎ

• Momentum conservation not required aspressure is assumed constant:

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DiffusionIIT Kanpur

, = , , + , ,ordinary diffusion due toconcentration gradient

thermal diffusion due totemperature gradient , = ,

, , = ∑mol fractionof species i

multicomponentdiffusion coefficient

not same as binarydiffusion coefficient

gradient ofspecies j

, = , , + , ,thermal diffusionvelocity

ordinary diffusionvelocity

• Multicomponent diffusion

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DiffusionIIT Kanpur = −

=matrix

= ∑ 1 − − 1 −number of species

binary diffusioncoefficient

, , = −thermal diffusioncoefficient

mass fractionof species i

• Thermal diffusion

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Boundary ConditionsIIT Kanpur

• Upstream: T(x -) = Tu and Yi(x -) = Yi,u

• Downstream: dT/dx (x +) = 0 and dYi/dx (x +) = 0

• is not known a priori and is part of the solution ( uVu = uSL)

• The domain: - < x < + is actually only ~cms

• Explicitly fix the coordinate system to move with the flame: T(x1) = T1

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Detailed Flame ModelIIT Kanpur

• Ideal-gas equation of state

• Relations for diffusion velocities (vi,diff)

• Temperature dependent species properties: hi(T), cp,i(T), ki(T) & DBij(T)

• Detailed chemical kinetic mechanism to obtain the • Interconversion relations for Xis, Yis and [Ci]s

• Assumption: no radiation, no KE and PE terms, constant pressure

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ResultsIIT Kanpur

Methane-Air ( = 1)

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ResultsIIT Kanpur Methane-Air ( = 1)