The Shell autoignition model

A new mathematical formulation

S. S. Sazhin, E. M. Sazhina, M. R. Heikal, C. Marooney, S. V. Mikhalovsky

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

The equations of the Shell model are reexamined with a view to their more effective implementation into a computational fluid dynamics code. The simplification of the solution procedure without compromising accuracy is achieved by replacing time as an independent variable with the fuel depletion, which is the difference between the initial fuel concentration and the current one. All the other variables used in this model, including temperature, concentration of oxygen, radicals, intermediate and branching agents are expressed as functions of fuel depletion. Equations for the temperature and concentration of the intermediate agent are of the first order and allow analytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case the numerical solution of five coupled first-order ordinary differential equations is reduced to the solution of only two coupled first- order differential equations for the concentration of radicals and branching agent. It is possible to rearrange these equations even further so that the equation for the concentration of the radicals is uncoupled from the equation for the branching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution of the first-order ordinary differential equation for the concentration of the branching agent and the solution of the first-order differential equation for time are presented in the form of integrals containing the concentration of the radicals obtained earlier. This approach allows the central processing unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model considerably more effective.

Original languageEnglish
Pages (from-to)529-540
Number of pages12
JournalCombustion and Flame
Volume117
Issue number3
DOIs
Publication statusPublished - May 1999
Externally publishedYes

Fingerprint

spontaneous combustion
formulations
Ordinary differential equations
differential equations
Differential equations
Oxygen
depletion
Program processors
Reactive Oxygen Species
Computational fluid dynamics
Temperature
oxygen
computational fluid dynamics
simplification
central processing units
temperature

ASJC Scopus subject areas

  • Energy Engineering and Power Technology
  • Fuel Technology
  • Mechanical Engineering

Cite this

Sazhin, S. S., Sazhina, E. M., Heikal, M. R., Marooney, C., & Mikhalovsky, S. V. (1999). The Shell autoignition model: A new mathematical formulation. Combustion and Flame, 117(3), 529-540. https://doi.org/10.1016/S0010-2180(98)00072-8

The Shell autoignition model : A new mathematical formulation. / Sazhin, S. S.; Sazhina, E. M.; Heikal, M. R.; Marooney, C.; Mikhalovsky, S. V.

In: Combustion and Flame, Vol. 117, No. 3, 05.1999, p. 529-540.

Research output: Contribution to journalArticle

Sazhin, SS, Sazhina, EM, Heikal, MR, Marooney, C & Mikhalovsky, SV 1999, 'The Shell autoignition model: A new mathematical formulation', Combustion and Flame, vol. 117, no. 3, pp. 529-540. https://doi.org/10.1016/S0010-2180(98)00072-8
Sazhin SS, Sazhina EM, Heikal MR, Marooney C, Mikhalovsky SV. The Shell autoignition model: A new mathematical formulation. Combustion and Flame. 1999 May;117(3):529-540. https://doi.org/10.1016/S0010-2180(98)00072-8
Sazhin, S. S. ; Sazhina, E. M. ; Heikal, M. R. ; Marooney, C. ; Mikhalovsky, S. V. / The Shell autoignition model : A new mathematical formulation. In: Combustion and Flame. 1999 ; Vol. 117, No. 3. pp. 529-540.
@article{cdf41a109230475f8e012bf4539c8d8f,
title = "The Shell autoignition model: A new mathematical formulation",
abstract = "The equations of the Shell model are reexamined with a view to their more effective implementation into a computational fluid dynamics code. The simplification of the solution procedure without compromising accuracy is achieved by replacing time as an independent variable with the fuel depletion, which is the difference between the initial fuel concentration and the current one. All the other variables used in this model, including temperature, concentration of oxygen, radicals, intermediate and branching agents are expressed as functions of fuel depletion. Equations for the temperature and concentration of the intermediate agent are of the first order and allow analytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case the numerical solution of five coupled first-order ordinary differential equations is reduced to the solution of only two coupled first- order differential equations for the concentration of radicals and branching agent. It is possible to rearrange these equations even further so that the equation for the concentration of the radicals is uncoupled from the equation for the branching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution of the first-order ordinary differential equation for the concentration of the branching agent and the solution of the first-order differential equation for time are presented in the form of integrals containing the concentration of the radicals obtained earlier. This approach allows the central processing unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model considerably more effective.",
author = "Sazhin, {S. S.} and Sazhina, {E. M.} and Heikal, {M. R.} and C. Marooney and Mikhalovsky, {S. V.}",
year = "1999",
month = "5",
doi = "10.1016/S0010-2180(98)00072-8",
language = "English",
volume = "117",
pages = "529--540",
journal = "Combustion and Flame",
issn = "0010-2180",
publisher = "Elsevier",
number = "3",

}

TY - JOUR

T1 - The Shell autoignition model

T2 - A new mathematical formulation

AU - Sazhin, S. S.

AU - Sazhina, E. M.

AU - Heikal, M. R.

AU - Marooney, C.

AU - Mikhalovsky, S. V.

PY - 1999/5

Y1 - 1999/5

N2 - The equations of the Shell model are reexamined with a view to their more effective implementation into a computational fluid dynamics code. The simplification of the solution procedure without compromising accuracy is achieved by replacing time as an independent variable with the fuel depletion, which is the difference between the initial fuel concentration and the current one. All the other variables used in this model, including temperature, concentration of oxygen, radicals, intermediate and branching agents are expressed as functions of fuel depletion. Equations for the temperature and concentration of the intermediate agent are of the first order and allow analytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case the numerical solution of five coupled first-order ordinary differential equations is reduced to the solution of only two coupled first- order differential equations for the concentration of radicals and branching agent. It is possible to rearrange these equations even further so that the equation for the concentration of the radicals is uncoupled from the equation for the branching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution of the first-order ordinary differential equation for the concentration of the branching agent and the solution of the first-order differential equation for time are presented in the form of integrals containing the concentration of the radicals obtained earlier. This approach allows the central processing unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model considerably more effective.

AB - The equations of the Shell model are reexamined with a view to their more effective implementation into a computational fluid dynamics code. The simplification of the solution procedure without compromising accuracy is achieved by replacing time as an independent variable with the fuel depletion, which is the difference between the initial fuel concentration and the current one. All the other variables used in this model, including temperature, concentration of oxygen, radicals, intermediate and branching agents are expressed as functions of fuel depletion. Equations for the temperature and concentration of the intermediate agent are of the first order and allow analytical solutions. The concentrations of oxygen and fuel are related via an algebraic equation which is solved in a straightforward way. In this case the numerical solution of five coupled first-order ordinary differential equations is reduced to the solution of only two coupled first- order differential equations for the concentration of radicals and branching agent. It is possible to rearrange these equations even further so that the equation for the concentration of the radicals is uncoupled from the equation for the branching agent. In this case the equation for the concentration of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution of the first-order ordinary differential equation for the concentration of the branching agent and the solution of the first-order differential equation for time are presented in the form of integrals containing the concentration of the radicals obtained earlier. This approach allows the central processing unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model considerably more effective.

UR - http://www.scopus.com/inward/record.url?scp=0033136125&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033136125&partnerID=8YFLogxK

U2 - 10.1016/S0010-2180(98)00072-8

DO - 10.1016/S0010-2180(98)00072-8

M3 - Article

VL - 117

SP - 529

EP - 540

JO - Combustion and Flame

JF - Combustion and Flame

SN - 0010-2180

IS - 3

ER -