MATHEMATICAL MODELS OF FUEL SPRAY AUTOIGNITION

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MATHEMATICAL MODELS OF FUEL SPRAY AUTOIGNITION

• Vladimir Goldshtein

Ben-Gurion University of the Negev Department of
Mathematics
P.O.B. 653,
Beer-Sheva 84105 ISRAEL
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• IN COOPERATION WITH
• V.BIKOV
• J.B.GREENBERG
• I.GOLDFARB
• D.KATZ
• S.SAZHIN
• E.SAZHINA

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• Methodology.
• Physico-chemical processes with sufficiently
  different time scales are generally involved in
  spray combustion problems.
• Existence of a dispersion on time scale leads to
  specific type of a system of governing equations
  (singularly perturbed systems) and justifies an
  application of various asymptotical tools.
• A geometrical asymptotic method of integral
  manifolds, accepted by the authors, permits
  conduct analytical investigation of the
  considered systems.

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Invariant Manifolds Method
Semenov model of self-ignition
Initial conditions
Small parameters (due to the high activation
energy)
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Invariant Manifolds Method
- fast variable
- slow variable
The slow surface (curve in two-dimensional case)
Turning points
Fast motion (slow variable does not change)
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Trajectory Analysis
I1UO - slow trajectory, I1U is a fast part, UO is
a slow one (sitting on the attractive part of
the slow curve OTP). I3V - fast trajectory, it
does not approach the slow curve I2TPO - critical
trajectory, it moves via turning point T.
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Content

• Thermal explosion in mono-disperse sprays.
• Classification of main regimes.
• Long time delay before auto-ignition.
• Discussion on poly-disperse spray

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Thermal explosionin sprays
Main physical assumptions
- evaporating fuel droplets in combustible
gaseous mixture comprise a mono-disperse spray-
droplets are on the saturation line the mixture
is placed in a thermally insulated enclosure
(adiabatic approximation)- chemical reaction is
modeled as a one step highly exothermic first
order reaction
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MONODISPERSE SPRAY

• Energy and concentration equations
• Initial conditions

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Dimensionless System

• Dimensionless variables are introduced along the
  lines of classical Semenovs approach
• Dimensionless system reads as

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Dimensionless Parameters

• 3 dimensionless parameters describe competition
  between reaction and evaporation processes
• y - ratio of reaction heat to latent heat of
  evaporation
• e1 is defined by competition between
  combustion and evaporation.
• e2 represents the ratio between the potential
  energy of the combustible gas and the evaporation
  energy

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Slow Curve (zero approximation of invariant
manifold)

• The slow curve for the current
  system is given by the equation
• The shape and position of the slow curve
  in the plane depend on the
  combination of the five parameters of the system.
  Any combination of the parameters dictates the
  relative location of the slow curve to the
  initial point.
• Here q is a fast, and r is a slow variable.

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Possible Scenarios
Delayed trajectory ABTCS
Explosive trajectory DES
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Possible positions of the slow curve and
corresponding trajectories

• Delayed trajectory EFTG, the slow part FT belongs
  to the integral manifold.

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Summary of dynamical regimes

• Conventional fast explosion

Delayed explosion, concentration increases
Slow regime
Delayed explosion, concentration decreases
Freeze delay
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Delay Time (Case )

• The suggested type of analysis permits to
  evaluate the main characteristic times of the
  process tc1 - time of fast motion (evaporation
  and cooling), tc2 - time of slow motion

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Time HistoryThermal Explosion with Delay
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Theory versus Simulations

• Relative error () in the delay time upper bound
  versus the number of droplets per unit
  volume(a) n-decane fuel, Cf0 10-4 kmol/m3,
  Rd0 10-5 m(b) tetralin fuel, Cf0 10-4
  kmol/m3, Rd0 10-6 m.

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OXIDIZER

• Dimensionless system includes four ODEs and reads
  as
• Initial conditions

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Reduced System

• The linear integrals exist due to the adiabatic
  approach
• The number of the variables can be reduced

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Classification of Variables
Variables can be subdivided according to their
rates of change Case A.
radius is fast variable, temperature is
intermediate, and concentrations are slow
ones. Case B.
temperature is fast, radius and two
concentrations are slow.
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Slow Curves

• The slow curves of the considered system are
  determined by the equations
• Frank-Kamenetskii approximation (b0) is used.
• This simplifies sufficiently the slow curve
  equation, as well as the further system analysis
  on the slow curve.

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Case A Slow Curves and Trajectory Analysis

• The slow curve QOU of the considered system
  (radius is a fast variable).
• Part PO describes fast decrease of the droplets
  radius. At the point O the radius vanishes and
  the model is no longer valid.
• Part OU corresponds to conventional gaseous
  thermal explosion.

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Case B Slow Curves and Trajectory Analysis

• The slow curve QT2T1U of the considered system.
• Parts P1A and PB describe fast change of the gas
  temperature.
• Parts AT2 and BU represent slow motion along the
  slow curve.
• Turning point T2 serves as initial point of final
  thermal explosion.

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Criterion of Explosion

• A - parametric region of conventional thermal
  explosion (chemical reaction dominates the
  evaporation process, the system explodes).
• B - parametric region of delayed explosion
  (chemical reaction and evaporation are balanced
  on the slow curve).

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Evaporation Time
Case A. Evaporation time represents a time of the
fast motion from the initial point to the slow
curve (radius changes from the unity to zero).
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The Delay Time

• Case B. Delay time is defined as a time of
  trajectorys motion along the slow curve.
  Analytical estimations for the delay time are
  derived.

hypergeometric function of the
second kind
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Theory versus Numerics
Left - dimensionless delay time (103?delay)
versus dimensionless parameter ?1 for various
initial gas temperatures (red - 600 K, green-
650K, blue - 700K). Black circles - numerical
results, solid lines - theoretical predictions.
Right - corresponding relative error ().
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Theory versus Numerics(lean mixture)
Left - delay time (sec) versus initial
dimensionless fuel concentration for various
initial gas temperatures (red - 600 K, green-
650K, blue - 700K). Black circles - numerical
results, solid lines - theoretical predictions.
Right - corresponding relative error ().
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RADIATION,FAST GAS TEMPERATURE
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,
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• Following figure shows the dependence of the
  delay time on the droplets number for different
  values of droplet radii 1 (mkm), 1.1 (mkm), 1.2
  (mkm), 1.3 (mkm).

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CONCLUSIONS FOR MONODISPERCE SPRAY

• Possible types of dynamical behaviour of the
  system are classified and parametric regions of
  their existence are determined analytically.
• It is demonstrated that the original problem can
  be decomposed into two sub problems, due to the
  underlying hierarchical timescale structure.
• The first sub problem relates to the droplet heat
  up period, for which a relatively rapid time
  scale is applicable. The second subproblem begins
  at the saturation point.
• For the latter more significant second stage, it
  is found that there are five main dynamical
  regimes
• slow regimes, conventional fast explosive
  regimes, fast explosion regimes with two
  different types of fast explosion with delay
  (the concentration of the combustible gas
  decreases or increases).

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POLY-DISPERSE SPRAY

• Dimensional System

Energy equation
Mass equations
Concentration equation
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Problem formulation

• Non-dimensional variables (Semenov type)

Dimensionless system