Untitled - Aerobib - Universidad Politécnica de Madrid

7.2. TURBULENT COMBUSTION THEORIES 211
be the laminar and turbulent thermal diffusivities, respectively. The turbulent propagation
velocity u t of the flame is obtained from u l when x is substituted by ε. Since
u l is proportional to √ x, it results for u t
u t
u l
=
√ ε
x
(7.4)
Shelkin [14] also considers the influence of laminar diffusivity on the turbulent flame.
In this case, ε must be substituted in (7.3) by x + ε, thus resulting
√
u t
= 1 + ε u l x
(7.5)
This expression has the advantage over (7.4) that when turbulence approaches zero,
u t approaches u l .
Case 1) very seldom arises since under normal conditions the thickness of the
laminar flame is only of some tenths of a millimeter.
2) Scale of turbulence and flame thickness are of the same order of magnitude
l
d l
≃ 1. (7.6)
Damköhler does not consider this case. However Shelkin has proven that the
action of turbulence does not reduce then to an activation of the transport coefficients
but it also influences the combustion time of the mixture, which not only depends on
the reaction velocity of the species but on the rapidity of the turbulent mixing. To date,
no formula is available for this case.
3) Scale of turbulence is large when compared to flame thickness
l
d l
≫ 1. (7.7)
In this case combustion takes place through a laminar front but turbulence wrinkles
the laminar flame front thus increasing its surface. Thereby, the effective combustion
surface of the mixture is increased. Two possibilities should be considered
depending on the intensity of the turbulence.
3.a) The intensity of turbulence is very large with respect to the laminar propagation
velocity of the flame
v ′
u l
≫ 1. (7.8)
This case was not analyzed by Damköhler. The following consideration are
owned to Shelkin. When condition (7.8) takes place the structure of the combustion