Simplifying Chemical Kinetics

Transcription

1 Texas A&M University September 1, 2010

2 Chemically Reactive Flows: In gas-phase chemical reaction systems, the governing porcesses such as flow, chemical reactions and molecular transport, occur at time differ by order of magnitude.

3 Chemically Reactive Flows: In gas-phase chemical reaction systems, the governing porcesses such as flow, chemical reactions and molecular transport, occur at time differ by order of magnitude. In a typical spectrum of time scales, we can see that the chemical time scales cover a greater range than that of the physical time scales.

4 Thermo-chemistry vs flow time-scales

5 Homogeneous Well-Stirred Reactor Our chemical model is the following: ρ df α dt = ω α ns dt ρc p dt = P = ρr u T 1 ns f α M 1 α h α ω α

6 Homogeneous Well-Stirred Reactor Our chemical model is the following: ρ df α dt = ω α ns dt ρc p dt = P = ρr u T 1 ns f α M 1 α h α ω α where f α, ρ and P are the mass fraction, the total density and the pressure respectively.

7 Homogeneous Well-Stirred Reactor The prodcution rate of species i is given by where π j is given by nr ω i = M i j=1 π j = ( π j ρ α j A j T b j exp ns n=1 ( fn M n ) βnj E ) j R u T

8 Chemically Reactive Flows: The wide range of time scales in gas-phase chemical reaction systems is manifested as stiffness in the model system.

9 Chemically Reactive Flows: The wide range of time scales in gas-phase chemical reaction systems is manifested as stiffness in the model system. Such stiffness causes sloving the model system to be computationally infeasible.

10 Chemically Reactive Flows: The stiffness can be reduced by equilibrating the fast time scale processes and resolve only the slow time scale processes.

11 Chemically Reactive Flows: The stiffness can be reduced by equilibrating the fast time scale processes and resolve only the slow time scale processes. This reduction can be done by computing what is known as the slow manifold which is an invariant attracting manifold.

12 Differential Equations: We assume our model system of ODE s involves fast and slow dynamics. ẋ = f (x, y) ẏ = g(x, y) where x R m are the slow variables and y R n are the fast variables.

13 Invariant Manifold: We say that the curve y = h(x) is an invariant manifold if for an initial condition (x 0, y 0 ) such that y 0 = h(x 0 ), then the solution (x(t), y(t)) satifies y(t) = h(x(t)) for all t R. The stable manifold, unstable manifold, center manifold and the slow manifold are all invariant manifolds.

14 The Locally Linear Method: If the system has a slow manifold y = h(x), then dy dt = g(x, h(x)) = D xh(x) dx dt = D xh(x)f (x, h(x)

15 The Locally Linear Method: If the system has a slow manifold y = h(x), then dy dt = g(x, h(x)) = D xh(x) dx dt = D xh(x)f (x, h(x) If we differentiate g i with respect to x j and assuming local linearity we get k ( gi y ) k + g i = y k x j x j l ( yi x l f l x j )

16 The Locally Linear Method: If the system has a slow manifold y = h(x), then dy dt = g(x, h(x)) = D xh(x) dx dt = D xh(x)f (x, h(x) If we differentiate g i with respect to x j and assuming local linearity we get k ( gi y ) k + g i = y k x j x j l ( yi x l f l x j ) We use the above equation to estimating the Jacobian y i x j.

17 Maas-Pope vs LL Girimaji: The LL method of Girimaji is conceptually simpler than ILDM of Maas and Pope.

18 Maas-Pope vs LL Girimaji: The LL method of Girimaji is conceptually simpler than ILDM of Maas and Pope. In LL method, there is no need of computing eigensystems or martix decompositions as we do in ILDM.

19 Maas-Pope vs LL Girimaji: The LL method of Girimaji is conceptually simpler than ILDM of Maas and Pope. In LL method, there is no need of computing eigensystems or martix decompositions as we do in ILDM. We expect the LL method of Girimaji is computationally less expensive than ILDM of Maas and Pope.

20 Maas-Pope vs LL Girimaji:

21 Testing the methods:(davis and Skodje, 1999) We consider the following system ẋ = x (γ 1)x + γx 2 ẏ = γy (1 + x) 2 It has an equilibrium point (0, 0) and an invariant manifold defined by h(x) = x 1 + x.

22 Graph 1:

23 Maas-Pope vs LL Girimaji: 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.

24 Graph 2, Maas-Pope vs LL Girimaji:

25 Maas-Pope vs LL Girimaji: For any system of the form ẋ = f (x) ẏ = γy + g(x) It can be shown that Maas and Pope ILDM approximation and Girimaji locally linear method give the same slow manifold which is given by h(x) = 1 ( g(x) + g (x)f (x) ) γ γ + f (x)

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

27 Plan: Our goal in this project is the following:

28 Plan: Our goal in this project is the following: Writing a matlab code for the full HWS reactor.

29 Plan: 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.

30 Plan: 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.

31 Plan: 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 Maas and Pope.

32 Thank you: QUESTIONS?