Etude de la combustion de gaz de synthèse issus d'un processus de ...

Appendix C -Rivère model
Appendix C – RIVÈRE MODEL
The heat flux received by the wall is modelled using an original heat conduction
approach in the fluid-wall boundary based on the kinetic theory of gases developed by
Rivère, (2005), from Renault. The heat flux appears as a statistic result of the gas
molecules over the wall. Random reflection is assumed in direction, not in module.
Each gas molecule yields kinetic energy ΔE c when collides with the wall, characterized
by the kinetic energy fraction K:
' 2 '2
' 1
2 '2 Ec
− Ec
V −V
Δ Ec = Ec − Ec = mg
cos( θ )( V − V ) K = =
2
(C.1)
2
E V
With m g the molecular mass, θ the impact angle, V and V’ the impact and reflection
speeds (Figure C.1).
c
tel-00623090, version 1 - 13 Sep 2011
Figure C.1 – Fraction of energy received by the wall.
In order to know the kinetic energy fraction K yielded by one molecule of gas during
rebound, the wall is modelled by a linear chain of atoms vibrating by translation along x
only. The calculus shows that n=2 atoms are enough to take into account the
phenomenon (Figure C.2). The cases of harmonic and non-harmonic systems are
considered.
Figure C.2 – Molecules translation movement.
As a result of this calculation, the transfer coefficient K is obtained with error of about
2%. This coefficient is dependent of the gas temperature, which in turns determines the
thickness of the thermal boundary layer.
χ λ
K = η + −
T T
(C.2)
g
g
With η,λ, and χ the integration constants to be determined using experimental results.
220