
Research Article


Theoretical Notes on the Mathematical Modelling of Gaseous Detonations Using Boltzmann Equation 

K.M. Saqr,
N.A.C. Sidik
and
M.A. Wahid



ABSTRACT

In this study, we present some theoretical notes on the use of the Boltzmann equation in computing detonation of combustible gases. The scarceness of relevant study, as well as the sensitivity of the topic, were the major motivations behind the intention to present these notes for publication. A timeline of the detonation phenomenon and its physical comprehension was drawn to justify the unsuitability of the continuum hypothesis of describing gaseous detonations. Then, the theory behind the Boltzmann equation and DSMC is elaborated. Finally, a discussion of the available study on the topic is introduced and concluded.





Received: December 16, 2009;
Accepted: April 21, 2010;
Published: June 10, 2010


INTRODUCTION
This study outlines the past and contemporary efforts aim to achieve better
understanding of the detonation phenomena in gases. In fact, the majority of
these efforts are derived by the persisting demand to understand the detonation
wave front, which requires a molecular scope in analyzing the phenomena. In
addition, this article theoretically elucidates the physics of the Boltzmann
equation and its connection with DSMC method in the context of nonequilibrium
flows. The interest in investigating gaseous detonation come from its tremendous
energy conversion rate and the potential of employing the detonation phenomenon
in future hypersonic propulsion systems through pulse detonation engines (Saqr
et al., 2010) and in power generation systems (Wahid
et al., 2008a, b, 2009).
Another important motivation to use a molecular level methodology in simulating
gaseous detonations is the lack of comprehensive understanding of DDT (Deflagration
to Detonation Transition).
The detonation phenomenon was firstly observed as a violent chemical reaction
with the discovery and use of explosives in the fifteenth century. In fact,
the first explosive known to sustain detonation waves was the gold fulminate;
introduced by Oswald Croll in 1608 (Bacon and Rees, 2000).
However, detonation was not defined and distinguished apart from other forms
of combustion until the development of certain diagnostic tools which enabled
the measurement of the detonation wave velocity. This was probably done by Abel
(1869). The first efforts resulted in defining the range of detonation wave
velocity of several gaseous fuels and its dependency factors were revealed by
Berthelot and Vieille (1883). By the end of the nineteenth
century, detonation as a mode of combustion was clearly distinguished from deflagration
based on the propagation velocity. The chemical reaction in detonation waves
was reasoned to be initiated by the adiabatic compression in the detonation
front (Dixon, 1893).
Detonation can be defined as a shock wave sustained by a chemical reaction. However, to get deeper understanding of the nature and physics of detonation, sophisticated explanation of its mechanism is mandatory to comprehend. The chemical reaction associated with detonation consumes the combustible material about 10^{3} to 10^{8} faster than in other forms of combustion (i.e., deflagration). The meaning of this ultrahigh combustion speed can be appreciated if one compared the energy converted through detonation to a well know energy conversion reference. If a detonation process was initiated into a good solid explosive material, energy is converted at a rate of 10^{10} W cm^{2} of its detonation front. Two square meters detonation wave gives energy more than the total electric generating capacity of the United States in 2006, which was 1.075x10^{12} W.
This enormous energy conversion rate has motivated researchers since the early
days of the twentieth century to investigate the various aspects of detonation
theory and application. Since, the main property distinguishes detonation from
other forms of combustion is velocity, researchers fundamentally were interested
to calculate it. L. chapman and E. jouguet have formulated a theory
to predict the velocity of one dimensional detonation waves between 1899 and
1905 (Becker, 1922). In their study, Chapman and Jouguet
treated the detonation front as a discontinuity plane across which the conservation
waves of shock waves apply. In the CJ theory, the velocity of steady detonation
is consistent with the conservation conditions. Once the detonation wave is
known and the equation of state of the reaction products is given, the conservation
laws determine the final state behind the detonation front. This theory, however,
totally disregards the features of detonation structure because of the insinuation
of the one dimensional, adiabatic flow assumptions. Basically, it yields the
possible solution of the steady one dimensional conservation equations that
links the equilibrium states of the upstream reactants and downstream products.
In order to know the propagation mechanism of detonation waves, a more detailed
and generalized theory had to be introduced.
The ZND theory was named after Zildovich in 19141987, von Neumann in 19031957
and Döring in 19112006, who were conducting their research independently
in the 1940s. They based their study on the inviscid equations of Euler hydrodynamics.
This theory was the first to divide the detonation structure into a leading
shock front followed by a chemical reaction zone. Zel'dovich
(1940) investigated the effects of including the heat and momentum losses
on the detonation structure. This investigation aimed to rise above the assumptions
of the CJ theory. His study showed that at some critical values of the loss
terms, it was possible to explain the onset of detonation as the velocity is
much less than the equilibrium CJ value. In the same time, von Neumann managed
to demonstrate the pathological detonations which have velocities higher than
the CJ velocity. A year later, Döring studied the thermodynamic states
within the detonation zone. The ZND theory of detonation gave the most firm
explanation of the propagation mechanism in this furious form of combustion.
It proposes that in detonation, energy transfers by mass flow in strong compression
wave, on the contrary of deflagration, where the important energy transfer depends
on conduction. The detonation fronta shock wave is propagating with a supersonic
speed into the combustible medium causing its temperature to rise sharply. The
sharp temperature rise occurs due to the adiabatic compression effect of the
shock wave. Chemical reactions are being triggered simultaneously due to the
explosive nature of the combustible medium. Then, these chemical reactions supports
the propagation of the shock wave further into the medium (Fickett
and Davis, 1979).
One of the main significant contributions of the ZND theory is the presentation
of a thickness for the detonation front. This was achieved by series of the
Hugoniot curves representing the successive fractions of a reaction in detonation.
Later on, several researches have been conducted to evaluate the value of such
thickness under different conditions. Typically, this thickness equals two to
three meanfreepaths, which in return represent the characteristic length scale
of the detonation (O'Connor et al., 2006).
As a result, the Kn (Knudsen number) for detonations ranges between 0.3 to
0.5. This would take the detonation problem far beyond the limits of continuum
fluid mechanics, since that Kn limit for Euler and NavierStokes equations is
zero and 0.05, respectively (Long, 1991). For this reason,
in order to study the thermodynamic changes in the transition region of the
detonation front, it is necessary to use a governing equation that is valid
for Kn values greater than 0.05, which is able to model the macroscopic momentum
and heat fluxes within the detonation front. Boltzmann equation is the only
equation capable of achieving this goal since it is valid for all values of
Kn because of its molecular nature. Here, will be discuss the physics and advantages
of using Boltzmann equation in detonation problems as well as highlighting the
DSMC (Direct Simulation Monte Carlo) method to solve the Boltzmann equation.
THE BOLTZMANN EQUATION AND DSMC
Whenever Boltzmann equation is mentioned for gas dynamics problems, DSMC (Direct
Simulation Monte Carlo) method must come into consideration. Fundamentally speaking,
both the Boltzmann equation and DSMC are based on the kinetic theory of gases.
Classical texts explained this theory by giving the example of a billiards table,
where each ball represents a molecule of the gas and the cushions represent
the container (Jeans, 1925). Just the same as this mechanical
illustration, the kinetic theory assumes that all molecules have the same size
and are spherically identical. All molecules are constantly in motion, translating
and colliding with each others and with the cushions of a billiard table (or
the walls of a gas container). The kinetic theory describes the macroscopic
behavior of the system in respect to the microscopic motion of its molecules.
Temperature, for example, represent the intensity of random motion of the molecules,
while pressure represents the molecular force exerted on the container surfaces
(i.e., collision of the balls on the cushions.). It is vital in this sense to
comprehend this example and concepts in order to understand the physics behind
the Boltzmann equation and the DSMC method.
Microscopic interactions are accountable for all the phenomena that occur in
a gas flow. They govern the variation in properties across the fluid flow and
also govern the transfer of heat across surfaces as well as the generation of
forces on these surfaces. These variations in fluid properties are often formulated
in terms of statistical distributions, which are also used to formulate the
surface boundary condition. The Boltzmann equation arises from the attempt to
equate the variations in fluid properties to the behavior of the microscopic
interactions that generate these variations (Agbormbai, 2002;
Azwadi and Tanahashi, 2006, 2007).
The theory to connect between kinetic theory and fluid dynamics was commenced
by Hilbert (1912), Zeytounian (2002),
Azwadi and Tanahashi (2008) and Azwadi
et al. (2010). The famous mathematician Hilbert indicated how to
approximately solve the Boltzmann equation for the kinetic theory of gases in
1912. In the same year, Chapman and Enskog, independently devised approximate
solution of the Boltzmann equation valid for dense gases. Enskog followed a
method that generalizing Hilbert idea. His method was adopted by Chapman
and Cowling (1952) and Azwadi and Irwan (2010) and
became known as the ChapmanEnskog method.
In Bird (1963), expanded the Monte Carlo method into
gas dynamics and the method came to be known as Direct Simulation Monte Carlo,
or DSMC. The fundamental approach of this undeterministic simulation method
is to take statistical samples as the basis for predicting the physical behavior
of the fluid. In details, the DSMC method discretises the collision and convection
terms of the Boltzmann equation by calculating them separately within a time
interval that is small compared to the mean time between two collisions. Within
this small time interval, molecular interactions are decoupled from molecular
motions and within a discretised flow field molecules are moved, searched for
and allowed to collide. Associated colliding molecules are sampled from near
adjacents within the discrete cells. Molecules that are determined to have crossed
a surface boundary are made to undergo gassurface interactions, whereas those
that cross out of the flow field are abandoned Upstream, new molecules constantly
enter the flow field (Sone, 2007; Cercignani,
2000; Agbormbai, 2002).
The use of DSMC method with detonation problems started very recently in 2000
(Long and Anderson, 2000). The standard DSMC algorithm
consists of 5 steps, required for the simulation of gaseous detonations. Before
the simulation starts, initialization of the cells and molecules must be taken
over. A cell is a three dimensional computational domain in physical space that
will initially contain some random fraction of the total number of molecules
specified for the simulation. Normally, each cell has dimensions smaller than
the meanfreepath. The timestep size should be a small fraction of the mean
collision time between two molecules. After the initialization of the cells,
molecules are randomly introduced into the domain (i.e., cell) and the following
four uncoupled steps are repeated for every time step:
• 
Allow the molecules to move according to the timestep and
their velocity, while applying the boundary conditions 
• 
Organize the molecules into cells based on their new positions relative
to their local cell 
• 
Allow random collision between selected molecules 
• 
Provide the molecules with the opportunity (i.e., time) for chemical reaction
and/or redistribution of energy 
• 
After specified number of time steps, sample the microscopic properties
in each cell. Macroscopic flow properties are calculated after a specified
number of microscopic samples have been collected (O'Connor
et al., 2006) 
One major obstacle in developing codes using DSMC for gaseous detonations is
the computational power demand. Cell size would have to be infinitesimal in
order to be less than the mean free path. This implies that in order to investigate
detonation wave in a few millimeters size domain, the DSMC would have to run
over hundreds of thousands on cells. Keeping in mind the probabilistic nature
of DSMC, this would require lengthy hours to reach a steady detonation, even
with monatomic or diatomic gas mixtures.
A REVIEW OF LITERATURE
Research publications studying gaseous detonations using the Boltzmann equation
and DSMC are very scarce. A very important research was published in 2002 on
the pulse detonation engine. The research investigated twodimensional nanoscale
detonations which are dominated by wall effects (Genovesi
and Long, 2002). The researchers, who conducted this study in Pennsylvania
State University concluded that there is a relationship between the curvature
of the detonation wave and its detonation speed. However, they stated that the
existence of higher computational power to be utilized with the DSMC method
is the only way to investigate this relationship through multidimensional simulation
of detonations.
In 2004, a similar problem was simulated using DSMC (Sharma
and Long, 2004). The blast impact problem was formulated using an assumption
that the impact does not deform the container. The DSMC code was operated on
a parallel computing system. In the following year, another research paper was
published by the same group in Pennsylvania, U.S. The researchers studied the
ultra fast detonations which exceeds the steadystate velocities predicted by
the CJ and ZND theories (O'Connor et al., 2005).
Their results describe a case where the detonation front and the reaction zone
overlap, with an unpredictable behavior. However, the results obtained from
the DSMC simulation of such unusual detonations have not been verified to this
moment.
Two years later, a groundbreaking paper was presented in the Annual Joint
Propulsion Conference and Exhibit, U.S. The study analyzed and reported the
initial results of bimolecular detonations using DSMC in order to plot the thermodynamic
properties across the detonation wave front. This was successfully obtained,
while it was impossible using the continuum equations (i.e., Euler and/or NavierStokes).
The DSMC structure described in this paper was modified in order to comprise
more complex chemical reactions (O'Connor et al.,
2006).
CONCLUSIONS
The Boltzmann equation through DSMC is capable of studying gaseous detonations
at levels of detail that can not be reached by continuum approaches. The major
challenges in developing the DSMC codes for detonation problems are:
• 
The computing resources demand: which can be addressed through
the implementation of parallel computing facilities and algorithms and object
oriented programming approaches 
• 
The inclusion of complex chemical reaction models 
• 
Physical reasoning and fundamental explanation of the results obtained
across the detonation wave front, which can not be verified experimentally 

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