Karlovitz number

Physical key figure
Surname	Karlovitz number
Formula symbol	${\displaystyle {\mathit {Ka}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Ka}}={\frac {\tau _{\mathrm {L} }}{\tau _{\mathrm {K} }}}}$
${\displaystyle \tau _{\mathrm {L} }}$ Time scale for the spread of the laminar flame ${\displaystyle \tau _{\mathrm {K} }}$ smallest turbulent timescale
Named after	Béla Karlovitz
scope of application	turbulent combustion processes

The Karlovitz number is used to describe turbulent combustion processes and is composed of the ratio of the time scale for the propagation of the laminar flame to the smallest turbulent time scale ( Kolmogorov time ): ${\displaystyle \tau _{\mathrm {L} }}$ ${\displaystyle \tau _{\mathrm {K} }}$

${\displaystyle {\mathit {Ka}}={\frac {\tau _{\mathrm {L} }}{\tau _{\mathrm {K} }}}}$

The laminar flame time scale is usually defined as the time that the laminar flame front needs to move by flame propagation at the laminar flame speed by a distance that is the same as the laminar flame front thickness (including its preheating layer) . It can also be described by the ratio of the diffusion constant to the square of the laminar flame speed: ${\displaystyle \tau _{\mathrm {L} }}$ ${\displaystyle s_{\mathrm {L} }}$ ${\displaystyle l_{\mathrm {L} }}$ ${\displaystyle D}$

${\displaystyle \tau _{\mathrm {L} }={\frac {l_{\mathrm {L} }}{s_{\mathrm {L} }}}={\frac {D}{s_{\mathrm {L} }^{2}}}}$

If so, the heat and material diffusion within the flame front takes place much faster than all turbulence time scales. Thus, the local flame structure and the area of ​​the chemical reaction are not changed or influenced by turbulence and laminar conditions prevail within the flame . In this case, the flame can usually be described well with a flamelet approach, in which it is assumed that the flame front behaves completely laminar in a local approximation. ${\displaystyle {\mathit {Ka}}\ll 1}$

If so, the smallest turbulent eddies are equal to or smaller than the thickness of the preheating layer in the flame front. This can lead to a turbulent heat and material transport within the flame front. This leads to both a broadening of the flame front and an increase in the turbulent flame speed. ${\displaystyle {\mathit {Ka}}\geq 1}$

The naming of the dimensionless number refers to the Hungarian physicist Béla Karlovitz .

Connection with other quantities

According to the definition of the Karlovitz number above, the ratio can also be expressed using length scales or speeds:

${\displaystyle {\mathit {Ka}}={\frac {l_{\mathrm {L} }^{2}}{\lambda _{\mathrm {K} }^{2}}}={\frac {v_{\mathrm {K} }^{2}}{s_{\mathrm {L} }^{2}}}}$

Here stand for the Kolmogorov length (i.e. the smallest length scale that is influenced by the turbulence) and for the Kolmogorov velocity (i.e. the orbital velocity of eddies with the diameter of the Kolmogorov length). ${\displaystyle \lambda _{\mathrm {K} }}$ ${\displaystyle v_{\mathrm {K} }}$

Assuming that the Schmidt number applies, i.e. that the kinematic viscosity is roughly the same as the material diffusion constant , the Karlovitz number, the Damköhler number and the Reynolds number can be roughly related to each other as follows: ${\displaystyle {\mathit {Sc}}\approx 1}$ ${\displaystyle {\mathit {Da}}}$ ${\displaystyle {\mathit {Re}}}$

${\displaystyle {\mathit {Re}}\approx {\mathit {Da}}^{2}{\mathit {Ka}}^{2}}$

Alternative definition

If the laminar flame time scale is replaced by the reaction time scale in the above equation , a Karlovitz number can be defined for the influence of turbulence on the reaction layer: ${\displaystyle \tau _{\mathrm {R} }}$

${\displaystyle {\mathit {Ka}}_{\mathrm {R} }:={\frac {\tau _{\mathrm {R} }}{\tau _{\mathrm {K} }}}}$

Analogous to the definition of the laminar flame time scale, the reaction time scale describes the time that the flame front needs to cover a distance through flame propagation that is the same as the thickness of the reaction layer . The reaction layer is that section within the flame front in which the chemical reactions take place. In a laminar flame front, the reaction layer is considerably thinner than the preheating layer, which is characterized by material and heat diffusion. The size ratio is often described with a factor . So it applies ${\displaystyle l_{\mathrm {R} }}$ ${\displaystyle \delta \equiv {\frac {l_{\mathrm {R} }}{l_{\mathrm {L} }}}}$

${\displaystyle {\mathit {Ka}}_{\mathrm {R} }={\frac {l_{\mathrm {R} }^{2}}{\lambda _{\mathrm {K} }^{2}}}=\delta ^{2}{\mathit {Ka}}}$.

Typically a number is of the order of magnitude . ${\displaystyle \delta }$ ${\displaystyle O(10)}$

Analogous to the description above, it can be determined here:

If so, the chemical reactions take place much faster than any turbulence time scales. The internal structure of the reaction layer is thus not changed by turbulence, and laminar conditions prevail within the reaction layer. ${\displaystyle {\mathit {Ka}}_{\mathrm {R} }\ll 1}$

If so, the smallest turbulent eddies are equal to or smaller than the thickness of the reaction layer in the flame front. This can theoretically lead to a turbulent widening of the reaction layer. In extreme cases this would lead to a homogeneous distribution of the chemical reactions over a macroscopic volume (perfect stirred reactor). It is much more likely, however, that with such an intense turbulent disturbance of the reaction layer, local extinctions occur and the flame front breaks up or even goes out completely. ${\displaystyle {\mathit {Ka}}_{\mathrm {R} }\geq 1}$

Individual evidence

1. ↑ ^{a b c d} Norbert Peters: Turbulent Combustion . Cambridge University Press, 2000, pp. 78-79 .
2. ↑ Jürgen Warnatz, Ulrich Maas, Robert Dibble: Combustion, Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, 3rd Edition . Springer, 2001, p. 201-204 .
3. ↑ ^{a b} Chung K. Law: Combustion Physics . Cambridge University Press, 2006, pp. 496-500 .
4. ^ Bernard Lewis: Address by Dr. Bernard Lewis, Remarks on Combustion Science In: Symposium (International) on Combustion Volume 7 (Issue 1), 1958, pp. Xxxi – xxxv.