In
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combusti ...
, Frank-Kamenetskii theory explains the
thermal explosion of a homogeneous mixture of reactants, kept inside a closed vessel with constant temperature walls. It is named after a Russian scientist
David A. Frank-Kamenetskii
David Albertovich Frank-Kamenetskii (russian: Давид Альбертович Франк-Каменецкий, August 3, 1910 – June 2, 1970) was a Soviet theoretical physicist and chemist, professor and doctor of physical, chemical and mathe ...
, who along with
Nikolay Semenov developed the theory in the 1930s.
Problem description
Sources:
Consider a vessel maintained at a constant temperature
, containing a homogeneous reacting mixture. Let the characteristic size of the vessel be
. Since the mixture is homogeneous, the density
is constant. During the initial period of
ignition
Ignition may refer to:
Science and technology
* Firelighting, the human act of creating a fire for warmth, cooking and other uses
* Combustion, an exothermic chemical reaction between a fuel and an oxidant
* Fusion ignition, the point at which a ...
, the consumption of reactant concentration is negligible (see
and
below), thus the explosion is governed only by the energy equation. Assuming a one-step global reaction
, where
is the amount of heat released per unit mass of fuel consumed, and a reaction rate governed by
Arrhenius law
In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1 ...
, the energy equation becomes
:
where
Non-dimensionalization
Non-dimensional scales of time, temperature, length, and heat transfer may be defined as
:
where
;Note:
:''In a typical combustion process,
so that
.''
:''Therefore,
. That is, fuel consumption time is much longer than ignition time, so fuel consumption is essentially negligible in the study of ignition.''
:''This is why the fuel concentration is assumed to remain the initial fuel concentration
.''
Substituting the non-dimensional variables in the energy equation from the introduction
:
Since
, the exponential term can be linearized
, hence
:
At
, we have
and for
,
needs to satisfy
and
Semenov theory
Before
Frank-Kamenetskii, his doctoral advisor
Nikolay Semyonov (or Semenov) proposed a thermal explosion theory with a simpler model with which he assumed a linear function for the heat conduction process instead of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operator. Semenov's equation reads as
:
in which the exponential term
will tend to increase
as time proceeds whereas the linear term
will tend to decrease
. The relevant importance between the two terms are determined by the Damköhler number
. The numerical solution of the above equation for different values of
is shown in the figure.
Steady-state regime
When