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Numerical Algorithms for the Parametric Continuation of Stiff ODEs Deriving from the Modeling of Combustion with Detailed Chemical Mechanisms

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Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

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Abstract

The use of detailed chemical mechanisms is becoming increasingly necessary during the actual transition of energy production from fossil to renewable fuels. Indeed, the modern renewable fuels are characterized by a composition more complex than traditional fossil fuels due to the variability of the properties of the primary source, i.e. biomass. The parametric continuation can be a formidable tool to study the behavior of these new fuels allowing to promptly assess equilibrium conditions varying the main operative parameters. However, parametric continuation is a very computationally demanding procedure, both for the number of elementary operations needed and for the memory requirements. Actually, only very recently some approaches that allow affording this computation with chemical mechanisms composed of hundreds of chemical species and thousands of reactions have been proposed [1, 2, 37]. Starting from the procedure presented in [1], this paper illustrates further improvements of key steps that usually represents a bottleneck for the effective computation of parametric continuations and for the identification of bifurcation points.

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Abbreviations

\(\alpha \) :

continuation parameter

\(\lambda \) :

eigenvalues

\(\mathbf {f}\) :

right-hand side of system given by Eqs. (1–2)

\(\mathbf {J_f}\) :

Jacobian matrix

\(\mathbf {x}\) :

state vector

\(\psi \) :

test functions

\(\rho \) :

density, \(\text {kg}\,\text {m}^{-3}\)

\(\tau \) :

residence time, s

a :

real part of eigenvalues

b :

complex part of eigenvalues

\(c_p\) :

constant pressure specific heat, \(\mathrm{J\,kg}^{-1}\,\text {K}^{-1}\)

h :

mass specific enthalpy, \(\text {J}\,\text {kg}^{-1}\)

\(N_s\) :

number of chemical species

\(N_{nz}\) :

number of non zero element in a matrix

r :

net production rate, \(\text {kmol}\,\text {s}^{-1}\)

T :

temperature, K

t :

time, s

V :

volume of the reactor, \(\text {m}^{3}\)

W :

molecular weight, \(\text {kg}\,\text {kmol}^{-1}\)

Y :

mass fraction

F :

Fold Bifurcation

f :

feed conditions

H :

Hopf Bifurcation

j :

species index

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Authors and Affiliations

  1. Istituto di Ricerche sulla Combustione - CNR, Naples, Italy

    Luigi Acampora & Francesco S. Marra

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  1. Luigi Acampora
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  2. Francesco S. Marra
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Correspondence to Francesco S. Marra .

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  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Acampora, L., Marra, F.S. (2020). Numerical Algorithms for the Parametric Continuation of Stiff ODEs Deriving from the Modeling of Combustion with Detailed Chemical Mechanisms. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_1

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