simplifying chemical kinetics - aerospace...

Simplifying Chemical Kinetics

Ashraf Ibrahim

Texas A&M University

September 1, 2010

Ashraf Ibrahim Simplifying Chemical Kinetics

Chemically Reactive Flows:

In gas-phase chemical reaction systems, the governingporcesses such as flow, chemical reactions and moleculartransport, occur at time differ by order of magnitude.

In a typical spectrum of time scales, we can see that thechemical time scales cover a greater range than that of thephysical time scales.

In gas-phase chemical reaction systems, the governingporcesses such as flow, chemical reactions and moleculartransport, occur at time differ by order of magnitude.

In a typical spectrum of time scales, we can see that thechemical time scales cover a greater range than that of thephysical time scales.

Thermo-chemistry vs flow time-scales

Homogeneous Well-Stirred Reactor

Our chemical model is the following:

ρdfαdt

= ωα

ρCpdT

dt= −

ns∑1

hαωα

P = ρRuTns∑1

fαMα

where fα, ρ and P are the mass fraction, the total density andthe pressure respectively.

Our chemical model is the following:

ρdfαdt

= ωα

ρCpdT

dt= −

ns∑1

hαωα

P = ρRuTns∑1

fαMα

where fα, ρ and P are the mass fraction, the total density andthe pressure respectively.

The prodcution rate of species i is given by

ωi = Mi

nr∑j=1

πjραjAjT

bj exp(−

)where πj is given by

πj =ns∏n=1

( fnMn

The wide range of time scales in gas-phase chemical reactionsystems is manifested as stiffness in the model system.

Such stiffness causes sloving the model system to becomputationally infeasible.

The wide range of time scales in gas-phase chemical reactionsystems is manifested as stiffness in the model system.

Such stiffness causes sloving the model system to becomputationally infeasible.

The stiffness can be reduced by equilibrating the fast timescale processes and resolve only the slow time scale processes.

This reduction can be done by computing what is known asthe slow manifold which is an invariant attracting manifold.

The stiffness can be reduced by equilibrating the fast timescale processes and resolve only the slow time scale processes.

This reduction can be done by computing what is known asthe slow manifold which is an invariant attracting manifold.

Differential Equations:

We assume our model system of ODE’s involves fast and slowdynamics.

x = f (x, y)

y = g(x, y)

where x ∈ Rm are the slow variables and y ∈ Rn are the fastvariables.

Invariant Manifold:

We say that the curve y = h(x) is an invariant manifold if for aninitial condition (x0, y0) such that y0 = h(x0), then the solution(x(t), y(t)) satifies

y(t) = h(x(t))

for all t ∈ R. The stable manifold, unstable manifold, centermanifold and the slow manifold are all invariant manifolds.

The Locally Linear Method:

If the system has a slow manifold y = h(x), then

dt= g(x, h(x)) = Dxh(x)

dt= Dxh(x)f (x, h(x)

If we differentiate gi with respect to xj and assuming locallinearity we get∑

( ∂gi∂yk

∂yk∂xj

)+∂gi∂xj

(∂yi∂xl

∂fl∂xj

We use the above equation to estimating the Jacobian ∂yi∂xj

( ∂gi∂yk

∂yk∂xj

)+∂gi∂xj

(∂yi∂xl

∂fl∂xj

( ∂gi∂yk

∂yk∂xj

)+∂gi∂xj

(∂yi∂xl

∂fl∂xj

Maas-Pope vs LL Girimaji:

The LL method of Girimaji is conceptually simpler than ILDMof Maas and Pope.

In LL method, there is no need of computing eigensystems ormartix decompositions as we do in ILDM.

We expect the LL method of Girimaji is computationally lessexpensive than ILDM of Maas and Pope.

Testing the methods:(Davis and Skodje, 1999)

We consider the following system

x = −x

y = −γy − (γ − 1)x + γx2

(1 + x)2

It has an equilibrium point (0, 0) and an invariant manifold definedby

h(x) =x

1 + x.

Graph 1:

The following graph showing:1- the exact invariant manifold (solid black curve).2- the Maas and Pope/ LL Girimaji (dashed blue curve).3- GL Girimaji (dashed red curve).for γ = 5.

Graph 2, Maas-Pope vs LL Girimaji:

For any system of the form

x = f (x)

y = −γy + g(x)

It can be shown that Maas and Pope ILDM approximation andGirimaji locally linear method give the same slow manifold which isgiven by

h(x) =1

(g(x) +

g ′(x)f (x)

γ + f ′(x)

Graph 3, A chemical reaction (work with S. Suman):

Our goal in this project is the following:

Writing a matlab code for the full HWS reactor.

Writing a matlab code for LL Girimaji for the reactor.

Do the same as above in C++ or Fortran.

Show the equivalence between LL Girimaji and ILDM of Maasand Pope.

Thank you:

QUESTIONS ?

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