Kinetic approach to combustion processes in a recombination …calvino.polito.it/~mmkt/conforto.pdf · Kineticapproachtocombustionprocesses inarecombinationreaction FiammettaConforto

Kinetic approach to combustion processesin a recombination reaction

Fiammetta Conforto

Department of Mathematics – University of MessinaV.le F. Stagno d’Alcontres 31 – 98166 Messina – Italy

e-mail: [email protected]

VI Edition of the Summer School Methods & Models of Kinetic TheoryPorto Ercole, Italy, June 3 - 9, 2012

joint work withMaria Groppi and Giampiero Spiga, University of Parma, Italy

Roberto Monaco, Politecnico of Torino, ItalyAngela Ricciardello, University of Messina, Italy

Fiammetta Conforto (Messina) Steady Combustion Waves M&MKT12 1 / 53

The problemSteady 1–D combustion problems are investigated in a binary gas mixturemade up by atoms A, of mass m, and diatomic molecules A2, of mass 2m,undergoing an irreversible exothermic two–steps reaction through anunstable molecule Aexc

2 , with a very short mean lifetime

A + A→ Aexc2 → A2

The energy of chemical link of the molecule A2 is E0 > 0.The transition state Aexc

2 , endowed with a variable internal energy E > 0,gets de–excited to its ground state A2, supplying its excitation energyE + E0 to the gas mixture in the form of thermal energy.

I. Müller, in Asymptotic Methods in Nonlinear Wave Phenomena, World Scientific, 2007.

M. Bisi, M. Groppi, G. Spiga, Kinetic and Related Models, 3, (2010).F. C., M. Groppi, R. Monaco, G. Spiga, Kinetic and Related Models, 4, (2010).F. C., M. Groppi, R. Monaco, G. Spiga, in Proceedings WASCOM 2011, in press.


The Combustion Problem

Schematic diagram of a stationary one–dimensional combustion wave.

Unburned State (x → −∞)

c0 = 1, n0 > 0 , u0 > 0 , T0 > 0 , J0 = 0 , q0 = 0 ,Metastable Equilibrium State: A >> KT0, A activation energy

Burned State (x → +∞)

ceq = 0, neq > 0 , ueq > 0 , Teq > 0 , Jeq = 0 , qeq = 0 .Equilibrium State

K.K. Kuo, Principles of Combustion, John Wiley & Sons, 2005.


Common Assumption in Combustion Modelsreacting mixture can be treated as a continuum with ideal gas EOSsimple, one step, forward fast irreversible reactionLewis, Schmidt, and Prandtl numbers equal to oneequal mass diffusivities for all species and validity of Fick’s law ofdiffusionconstant specific heatsuniform pressure for low–speed combustion processDufour and Soret effects are negligible



The Combustion Problem - The Hugoniot Diagram

Hugoniot curve and Rayleigh lines on p versus 1/ρ plane.



The Governing Equations

Kinetic description The reactive gas is a mixture of three species, labeledby indices 1, 2, 3 for species A, A2, Aexc

2 respectively, described bydistribution functions f1(v), f2(v), ϕ3(v,E ), varying with time t andposition x. Reactive Boltzmann equations read as

∂f1∂t + v · ∂f1

∂x = Q11(f1, f2) +Q12(f1, f2) + J1(f1, f2, ϕ3) ,

∂f2∂t + v · ∂f2

∂x = Q21(f2, f1) +Q22(f2, f2) + J2(f1, f2, ϕ3) ,

∂ϕ3∂t + v · ∂ϕ3

∂x = J3(f1, f2, ϕ3) ,

where mechanical (elastic) collision integrals Q have standard forms andproperties, and chemical collision operators J , accounting forrecombination (superscript r) and inelastic scattering (superscript i) takethe form


J1 =

∫B i13(g ′ ,E )Π1,i

13 (v′ ;w′ ,E ;→ v)f1(v′)ϕ3(w′ ,E )dEdv′dw′

− f1(v)

∫B i13(g ,E )ϕ3(w,E )dEdw− 2f1(v)

∫Br11(g)f1(w)dw ,

J2 =

∫B i31(g ′ ,E )Π2,i

31 (v′ ,E ;w′ ;→ v)ϕ3(v′ ,E )f1(w′)dEdv′dw′

+ 2∫

B i23(g ′ ,E )Π2,i

23 (v′ ;w′ ,E ;→ v)f2(v′)ϕ3(v′ ,E )dEdv′dw′

− f2(v)

∫B i23(g ,E )ϕ3(w,E )dEdw ,

J3 =

∫Br11(g ′)Π3,r

11 (v′ ;w′ ;→ v,E )f1(v′)f1(w′)dv′dw′

− ϕ3(v,E )

∫f1(w)B i

31(g ,E )dw− ϕ3(v,E )

∫f2(w)B i

32(g ,E )dw .

M. Groppi, A. Rossani, G. Spiga, J. Phys. A: Math. Gen., 33 (2000).


Bαij denotes the microscopic collision frequency for a collision of type

α between species i and j ;g denotes relative speed |v−w|;Π2,i31 (v′ ,E ;w′ ;→ v) represents the probability density that the

outcoming particle 2 attains velocity v as a result of inelasticscattering of a particle 3 at velocity v′ and energy E with a particle 1at velocity w′ ;the recombination transition probability is completely determined bymomentum and energy conservations, and takes the form

Π3,r11 (v′ ;w′ ;→ v,E ) = δ

(12(v′ + w′)− v

)δ

(14m(v′ −w′)2 − E

)in terms of Dirac delta functions, from which one infers∫

dE∫

Π3,r11 (v′ ;w′ ;→ v,E )dv = 1 ,∫

dE∫

2vΠ3,r11 (v′ ;w′ ;→ v,E )dv = v′ + w′ ,∫

dE∫

(mv2 + E )Π3,r11 (v′ ,w′ ;→ v,E )dv =

12mv ′2 +

12mw ′2 ;


chemical conservation laws are expressed by∫J1(v)dv + 2

∫J2(v)dv + 2

∫J3(v,E )dvdE = 0 ,∫

vJ1(v)dv + 2∫vJ2(v)dv + 2

∫vJ3(v,E )dvdE = 0 ,∫ 1

2mv2J1(v)dv +

∫(mv2 − E0)J2(v)dv

+

∫(mv2 + E )J3(v,E )dvdE = 0 ,

to be combined with the well known conservation properties of theelastic collision operators.


The sequence (m, 2m, 2m) yields the continuity equation

∂ρ

∂t +∇ · (ρu) = 0 .

The sequence (mv, 2mv, 2mv) yields the momentum conservation equation

∂

∂t (ρu) +∇ · (ρu⊗ u + P) = 0 .

The sequence (mv2/2,mv2 − E0,mv2 + E ) yields the energy conservationequation

∂

∂t (E + Ech) +∇ · [(E + Ech)u + P · u + q + qch] = 0 ,

where

E =12ρu2 +

32nKT , Ech = −E0n2 +

∫Eϕ3(v,E )dEdv ,

qch = −E0n2(u2 − u) +

∫(v− u)Eϕ3(v,E )dEdv .


The number density of molecules is N = n2 + n3, the number density ofparticles is n = n1 + N, and the mass density is given by ρ = m(n1 + 2N).The number of atoms is not preserved by reactions, and in fact the weakform corresponding to the sequence (1, 0, 0) is not a conservation, butreads

∂n1∂t +∇ · (n1u1) = −2S , S =

∫Br11(g)f1(v)f1(w)dvdw .

It is easy to show that the following state

f ∗1 (v) = 0 , f ∗

2 (v) = N∗M2(|v− u∗|,KT ∗) , ϕ∗3(v,E ) = 0 ,

whereMi (v , θ) =

( mi2πθ

)3/2exp

(−mi2θ v2

)is a normalized Maxwellian, is collision equilibrium for the kineticequations, with five free parameters N∗ (density of molecules), u∗ (massvelocity of the gas), and T ∗ (kinetic temperature of the gas).


For a model of inverse power intermolecular potential

Br11(g) = kgαH

(g2 − 4A

m

),

with strength k and with an activation energy A accounted for by theHeavyside function H, an easy calculation provides, in dimensional form,

S = kn21(4KT

m

)α/2 2√π

Γ

(α + 32 ,

AKT

),

where Γ denotes incomplete Euler gamma function.The Fourier and Fick’s laws for a binary mixture of hard spheres

q(1) = −λ√

T ∇T , u(1)1 − u(1)2 = −D12n√

Tn1n2

∇c1 ,

with the constraintn2u(1)2 = −(1/2)n1u(1)1 .

S. Takata, K. Aoki, Phys. Fluids, 11 (1999).S. Takata, Phys. Fluids, 16 (2004).


We are interested here in the physical regime in which the process is drivenby elastic and inelastic scattering, with slow recombination reaction,namely mechanical and de–excitation times are much shorter than themacroscopic scale, whereas the reactive recombination time is muchlonger. In this regime, a suitable non–dimensionalization leads then to thescaled equations

∂f1∂t + v · ∂f1

∂x =1ε

(Q11 +Q12) + εJ r1 +

1εJ i1 ,

∂f2∂t + v · ∂f2

∂x =1ε

(Q21 +Q22) +1εJ i2 ,

∂ϕ3∂t + v · ∂ϕ3

∂x = εJ r3 +

1εJ i3 ,

where ε is a small parameter, playing the role of the classical Knudsennumber.


The asymptotic limit for ε→ 0 to first order accuracy is deduced byperforming a Chapman–Enskog analysis up to the Navier–Stokes level.Proceeding with the first order expansion for distribution functionsfi = f (0)i + ε f (1)i , i = 1, 2, ϕ3 = ϕ

(0)3 + εϕ

(1)3 , selecting n, n1, u, and T as

hydrodynamic fields, the reactive Navier–Stokes equations are obtained

∂n1∂t +∇ · (n1u) + ε∇ ·

(n1u(1)1

)= −2ε S ,

∂

∂t (2n − n1) +∇ · [(2n − n1)u] = 0 ,

∂

∂t [m(2n − n1)u] +∇ · [m(2n − n1)u⊗ u] +∇(nT ) + ε∇ · P(1) = 0 ,

∂

∂t

[m2 (2n − n1)u2 +

32nT − E0(n − n1)

]+∇ ·

[m2 (2n − n1)u2u +

52nTu− E0(n − n1)u

]+ε∇ ·

[P(1) · u + q(1) +

12E0n1u(1)1

]= 0 .


The Steady 1–D Navier–Stokes EquationsThe 1–D governing equations for the state variables c, n, u, T , J , q read

ddx (cnu + J) = − 4k√

πc2n2 Γ

(32 ,

AKT

),

ddx [(2− c)nu] = 0 ,

ddx(m(2− c)nu2 + nKT

)= 0 ,

ddx

[12m(2− c)nu3 +

52nKTu − E0(1− c)nu +

E02 J + q

]= 0 ,

J = − 2D122− c

√T dc

dx ,

q = −λ√

T dTdx ,

where c =n1n , ρ = m(2− c)n, u =

n1u1 + 2n2u2n1 + 2n2

, J = n1u(1)1 , q = q(1).


The Steady 1–D Combustion WaveIt is worth to introduce dimensionless field variables

n =nn0, u =

uu0, T =

TT0

, J =J

n0u0, q =

qmn0u3

0,

withc0 = n0 = u0 = T0 = 1 , J0 = q0 = 0 ,

ceq = Jeq = qeq = 0 , neq > 0 , ueq > 0 , Teq > 0 .Moreover,

x = x K n0u0λ√

T0, E0 =

E0KT0

, Tact =A

KT0,

M2 =u2

c2s, c2s =

5p3ρ =

5KT3m(2− c)

, M20 =

3mu20

5KT0,

L =λ

KD12, µ =

λk√

T0Ku2

0.


The Steady 1–D Combustion Wave

From the conservation equations, n, T and q are expressed in terms of c,u and J as follows

n =1

(2− c)u ,

T = (2− c)

[(1 +

53M2

0

)u − 5

3M20u2

],

q =12 +

32M2

0+ 2u2 −

(52 +

32M2

0

)u +

3E5M2

0

(1− c2− c −

J2

).

Let us note thatn > 0 ⇒ u > 0

andT > 0 ⇒ 0 < u < umax := 1 +

35M2

0.


From the remaining equations, we obtain

dcdx = −L

2 (2− c)J√T,

dudx = (2− c)−1

(1 +

53M2

0 −103 M2

0u)−1 1√

T

{−E0

1− c2− c

+J2

[E0 − L (2− c)

(1 +

53M2

0 −53M2

0u)

u]

+56M2

0

(u2 − 1

)+

52 (u − 1)

(1− 5

3M20u)}

,

dJdx = L J

(2− c)√

T− 4µ√

π

c2(2− c)2u2 Γ

(32 ,

TactT

).

Notice that the system admits the following critical value

ucrit :=12

(1 +

35M2

0

)=

12umax .


The dynamical system admits the following two equilibria

u±eq =

38M2

0

1 +53M2

0 ±

√(M2

0 − 1)2 − 16

15E0M20

, ceq = 0 , Jeq = 0 ,

which are real only if the parameter M0 satisfies the inequalitiesM2

0 ≤ M−eq < 1 or M2

0 ≥ M+eq > 1 ,

where

M±eq = 1 +

815E0 ±

√64225E 2

0 +1615E0 .

When M20 = M−

eq or M20 = M+

eq , a unique equilibrium stateueq = us , ceq = 0, Jeq = 0,

is obtained. MoreoverM2 < 1 ⇐⇒ u < us ,

whereus =

58 +

38M2

0=

58umax =

54ucrit .


The Steady 1–D Combustion Wave - Stability

For what concerns the stability analysis of the dynamical system, theassociated Jacobian matrix in the equilibrium state (0, u±

eq, 0) exhibitsthree real eigenvalues given by

λ1 = 0, λ2 =L

2√

T ±eq, λ3 = ∓

5√

(M20 − 1)2 − 16

15E0M20

4(1 + 5

3M20 − 10

3 M20u±

eq)√

T ±eq,

whereT ±

eq = 2(1 +

53M2

0 −53M2

0u±eq

)u±

eq

is the dimensionless temperature evaluated at the equilibrium (0, u±eq, 0).

The sign of the third eigenvalue depends on

u±eq < ucrit or u±

eq > ucrit .


The Combustion Problem - The Hugoniot Diagram

From conservation equations, we get the dimensionless Rayleigh line

p − 1 = −53M2

0 (v − 1) ,

and the Hugoniot hyperbola(p +

14

)(v − 1

4

)=

Q2 +

1516 ,

withQ(c, J , q) = E0

1− c2− c −

12E0J −

53M2

0q .

Q is varying with xQ ∈

[Q0 = 0,Qeq =

E02

].


The Combustion Problem - The Hugoniot DiagramThe Rayleigh line and the Hugoniot curve intersect at two points as long as

Q < Qcr , Qcr =1532

(M20 − 1)2

M20

.

The threshold of this range may be seen as the condition ensuring that

Qeq =E02 ≤ Qcr ⇐⇒ M2

0 ≤ M−eq , or M2

0 ≥ M+eq ;

moreover, for Q = Qeq = E0/2, ML ≡ M−eq and MU ≡ M+

eq.


The Steady 1–D Combustion Wave

u−eq ≤ us ≤ u+

eq ⇔ M20 ≤ M−

eq or M20 ≥ M+

eq, ∀E0 > 0ucrit < us < umax , ∀M2

0 > 0, ∀E0 > 0u+

eq < umax ⇔ M20 >

35(3+E0)

:= M1, ∀E0 > 0

u+eq > ucrit , ∀M2

0 > 0, ∀E0 > 0u−

eq < ucrit ⇔ M20 < M−

2 < 1 or M20 > M+

2 > 1, ∀E0 > 0where

M±2 =

35

(2 + E0 ±

√1 + 4E0 + E 2

0

)ucrit < 1 ⇔ M2

0 > 3/5, ∀E0 > 0us < 1 ⇔ M2

0 > 1, ∀E0 > 0u±

eq < 1, ∀M20 > 1, ∀E0 > 0

M1 < M−2 < M−

eq < 1 and 1 < M+eq < M+

2 , ∀E0 > 0


The Steady 1–D Deflagration Wave

Deflagration waves ⇔ M20 ≤ M−

eq, with E0 >14

In this regime, M1 < M−2 < M+

eq < 3/5 < 1.1a. if M2

0 ≤ M1, then1 < u−

eq < ucrit < us < umax ≤ u+eq

weak deflagration from (1, 1, 0) to (0, u−eq, 0), λ3 > 0

1b. if M1 < M20 ≤ M−

2 , then1 < u−

eq ≤ ucrit < us < u+eq < umax

weak deflagration from (1, 1, 0) to (0, u−eq, 0), λ3 > 0

discontinuous strong deflagration from (1, 1, 0) to (0, u+eq, 0), λ3 > 0

1c. if M−2 < M2

0 ≤ M−eq, then

1 < ucrit < u−eq ≤ us ≤ u+

eq < umax

discontinuous weak deflagration from (1, 1, 0) to (0, u−eq, 0), λ3 < 0



The Steady 1–D Weak Deflagration WaveThis simulation has been done by setting

Tact = 25 , E0 = 10 , L = 1 , M20 = 0.01 ,

for which ceq = 0, Jeq = 0, u−eq = 3.078, (T −

eq = 5.941), µ = 3695.

0 10 20 30 40 50 60 70 80

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Q=1/2 E0

Q=0

Qcr

ueq+

ueq−

*

Hugoniot diagram


Continuous Weak Deflagration Wave - trend of c

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

cHxL

E0=15, Μ=1449

E0=10, Μ=3700

E0=5, Μ=20500

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

cHxL

E0=15, Μ=28000

E0=10, Μ=3000

E0=5, Μ=6208

Comparison of c in the regimes 1a. M20 < M1 (up) and 1b. M1 < M2

0 < M−2 (down)varying E0, with L = 1, Tact = 25, M2

0 = 0.01 (up) and M20 = 0.05 (down)


Continuous Weak Deflagration Wave - trend of u

200 400 600 800 1000x

1

2

3

4

uHxL

E0=15, Μ=1449

E0=10, Μ=3700

E0=5, Μ=20500

200 400 600 800 1000x

1

2

3

4

5

uHxL

E0=15, Μ=28000

E0=10, Μ=3000

E0=5, Μ=6208

Comparison of u in the regimes 1a. M20 < M1 (up) and 1b. M1 < M2


0 = 0.01 (up) and M20 = 0.05 (down)


Continuous Weak Deflagration Wave - trend of T

200 400 600 800 1000x

2

4

6

8

T

E0=15, Μ=1449

E0=10, Μ=3700

E0=5, Μ=20500

200 400 600 800 1000x

1

2

3

4

5

6

7

T

E0=15, Μ=3000

E0=10, Μ=6208

E0=5, Μ=28000

Comparison of T in the regimes 1a. M20 < M1 (up) and 1b. M1 < M2


0 = 0.01 (up) and M20 = 0.05 (down)


Continuous Weak Deflagration Wave - trend of p

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

E0=15, Μ=1449

E0=10, Μ=3700

E0=5, Μ=20500

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

E0=15, Μ=3000

E0=10, Μ=6208

E0=5, Μ=28000

Comparison of p in the regimes 1a. M20 < M1 (up) and 1b. M1 < M2


0 = 0.01 (up) and M20 = 0.05 (down)


Continuous Weak Deflagration Wave - trend of J

200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

JHxL

E0=15, Μ=1449

E0=10, Μ=3700

E0=5, Μ=20500

200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

JHxL

E0=15, Μ=28000

E0=10, Μ=3000

E0=5, Μ=6208

Comparison of J in the regimes 1a. M20 < M1 (up) and 1b. M1 < M2


0 = 0.01 (up) and M20 = 0.05 (down)



200 400 600 800 1000x

0.5

1.0

1.5

2.0

2.5

3.0

3.5

uHxL

M02

=0.046, Μ=80000

M02

=0.01, Μ=44500

M02

=0.001, Μ=40000

200 400 600 800 1000x

1

2

3

4

uHxL

M02

=0.075, Μ=10500

M02

=0.06, Μ=7400

M02

=0.047, Μ=5900


0 < M−2 (down)varying M2

0 , with L = 1, Tact = 35, E0 = 10Fiammetta Conforto (Messina) Steady Combustion Waves M&MKT12 31 / 53


200 400 600 800 1000x

1

2

3

4

5

6

T

M02

=0.046, Μ=80000

M02

=0.01, Μ=44500

M02

=0.001, Μ=40000

200 400 600 800 1000x

1

2

3

4

5

T

M02

=0.075, Μ=10500

M02

=0.06, Μ=7400

M02

=0.047, Μ=5900





200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

M02

=0.046, Μ=80000

M02

=0.01, Μ=44500

M02

=0.001, Μ=40000

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

M02

=0.075, Μ=10500

M02

=0.06, Μ=7400

M02

=0.047, Μ=5900





200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

0.6

JHxL

M02

=0.046, Μ=80000

M02

=0.01, Μ=44500

M02

=0.001, Μ=40000

200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

JHxL

M02

=0.075, Μ=10500

M02

=0.06, Μ=7400

M02

=0.047, Μ=5900





200 400 600 800 1000x

0.5

1.0

1.5

2.0

2.5

3.0

uHxL

Tact=35, Μ=44000

Tact=30, Μ=13400

Tact=25, Μ=3700

200 400 600 800 1000x

0.5

1.0

1.5

2.0

2.5

3.0

3.5

uHxL

Tact=35, Μ=88000

Tact=30, Μ=24500

Tact=25, Μ=6208


0 < M−2 (down)varying Tact , with L = 1, E0 = 10, M2

0 = 0.01 (up) and M20 = 0.05 (down)



200 400 600 800 1000x

1

2

3

4

5

6

T

Tact=35, Μ=44000

Tact=30, Μ=13400

Tact=25, Μ=3700

200 400 600 800 1000x

1

2

3

4

5

T

Tact=35, Μ=88000

Tact=30, Μ=24500

Tact=25, Μ=6208



0 = 0.01 (up) and M20 = 0.05 (down)



200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

Tact=35, Μ=44000

Tact=30, Μ=13400

Tact=25, Μ=3700

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

p

Tact=35, Μ=88000

Tact=30, Μ=24500

Tact=25, Μ=6208



0 = 0.01 (up) and M20 = 0.05 (down)



200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

0.6

JHxL

Tact=35, Μ=44000

Tact=30, Μ=13400

Tact=25, Μ=3700

200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5

0.6

JHxL

Tact=35, Μ=88000

Tact=30, Μ=24500

Tact=25, Μ=6208



0 = 0.01 (up) and M20 = 0.05 (down)


The Steady 1–D Deflagration Wave

Deflagration waves ⇔ M20 ≤ M−

eq, with 0 < E0 ≤14

In this regime, M1 < M−2 < 3/5 ≤ M+

eq < 1.2c. if M−

2 < M20 ≤ 3/5, then

1 ≤ ucrit < u−eq < us < u+

eq < umax

discontinuous weak deflagration from (1, 1, 0) to (0, u−eq, 0), λ3 < 0


2d. if 3/5 < M20 ≤ M−

eq, then

ucrit < 1 < u−eq ≤ us ≤ u+

eq < umax

weak deflagration from (1, 1, 0) to (0, u−eq, 0), λ3 < 0

strong deflagration from (1, 1, 0) to (0, u+eq, 0), λ3 > 0


The Steady 1–D Detonation Wave

Detonation waves ⇔ M20 ≥ M+

eqIn this regime, 1 < M+

eq < M+2 , ∀E0 > 0.

a. if M+eq ≤ M2

0 < M+2 , then

ucrit < u−eq ≤ us ≤ u+

eq < 1 < umax

weak detonation solutions connecting (1, 1, 0) to (0, u+eq, 0);

strong detonation solutions connecting (1, 1, 0) to (0, u−eq, 0);

b. if M20 ≥ M+

2 , then

u−eq ≤ ucrit < us < u+

eq < 1 < umax

weak detonation solutions connecting (1, 1, 0) to (0, u+eq, 0);

discontinuous strong detonation solutions connecting (1, 1, 0) to(0, u−

eq, 0)


The Steady 1–D Weak Detonation WaveThis simulation has been done by setting

Tact = 25 , E0 = 10 , L = 1 , M20 = 16 ,

for which ceq = 0, Jeq = 0, u+eq = 0.8212, (T+

eq = 9.473), µ = 20.95.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

2

4

6

8

10

12

14

16

18

20

v

p

Q=Qs

ueq−

ueq+

CJU

Q=0

Q=E0/ 2

Hugoniot diagram.


Continuous Weak Detonation Wave - trend of u

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

E0=16, Μ=3.65

E0=15, Μ=5.05

E0=14, Μ=6.85

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

E0=15, Μ=6

E0=10, Μ=27

E0=5, Μ=350

Comparison of u in the regimes a. M+eq < M2

0 < M+2 (up) and b. M2

0 > M+2 (down)

varying E0, with L = 1, Tact = 25, M20 = 19.1 (up) and M2

0 = 21 (down)Fiammetta Conforto (Messina) Steady Combustion Waves M&MKT12 42 / 53

Continuous Weak Detonation Wave - trend of T

200 400 600 800 1000x

5

10

15

T

E0=16, Μ=3.65

E0=15, Μ=5.05

E0=14, Μ=6.85

200 400 600 800 1000x

2

4

6

8

10

12

T

E0=15, Μ=6

E0=10, Μ=27

E0=5, Μ=350

Comparison of T in the regimes a. M+eq < M2

0 < M+2 (up) and b. M2

0 > M+2 (down)



Continuous Weak Detonation Wave - trend of p

200 400 600 800 1000x

2

4

6

8

10

p

E0=16, Μ=3.65

E0=15, Μ=5.05

E0=14, Μ=6.85

200 400 600 800 1000x

2

4

6

8

p

E0=15, Μ=6

E0=10, Μ=27

E0=5, Μ=350

Comparison of p in the regimes a. M+eq < M2

0 < M+2 (up) and b. M2

0 > M+2 (down)



Continuous Weak Detonation Wave - trend of J

200 400 600 800 1000x

0.1

0.2

0.3

0.4

JHxL

E0=16, Μ=3.65

E0=15, Μ=5.05

E0=14, Μ=6.85

200 400 600 800 1000x

0.1

0.2

0.3

0.4

0.5JHxL

E0=15, Μ=6

E0=10, Μ=27

E0=5, Μ=350

Comparison of J in the regimes a. M+eq < M2

0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

M02

=20, Μ=5.6

M02

=19, Μ=5.05

M02

=18, Μ=4.25

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

M02

=30, Μ=32

M02

=21, Μ=26.85

M02

=16, Μ=20.95


0 < M+2 (up) and b. M2

0 > M+2 (down)

varying M20 , with L = 1, Tact = 25, E0 = 15



200 400 600 800 1000x

2

4

6

8

10

12

14

T

M02

=20, Μ=5.6

M02

=19, Μ=5.05

M02

=18, Μ=4.25

200 400 600 800 1000x

2

4

6

8

T

M02

=30, Μ=32

M02

=21, Μ=26.85

M02

=16, Μ=20.95


0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

2

4

6

8

10

p

M02

=20, Μ=5.6

M02

=19, Μ=5.05

M02

=18, Μ=4.25

200 400 600 800 1000x

1

2

3

4

5

p

M02

=30, Μ=32

M02

=21, Μ=26.85

M02

=16, Μ=20.95


0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

0.1

0.2

0.3

0.4

JHxL

M02

=20, Μ=5.6

M02

=19, Μ=5.05

M02

=18, Μ=4.25

200 400 600 800 1000x

0.1

0.2

0.3

0.4

JHxL

M02

=30, Μ=32

M02

=21, Μ=26.85

M02

=16, Μ=20.95


0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

Tact=30, Μ=8.95

Tact=25, Μ=5

Tact=20, Μ=2.65

200 400 600 800 1000x

0.2

0.4

0.6

0.8

1.0

uHxL

Tact=30, Μ=60.5

Tact=25, Μ=26.9

Tact=20, Μ=11.3


0 < M+2 (up) and b. M2

0 > M+2 (down)

varying Tact , with L = 1, E0 = 15, M20 = 19 (up) and M2



200 400 600 800 1000x

2

4

6

8

10

12

14

T

Tact=30, Μ=8.95

Tact=25, Μ=5

Tact=20, Μ=2.65

200 400 600 800 1000x

2

4

6

8

T

Tact=30, Μ=60.5

Tact=25, Μ=26.9

Tact=20, Μ=11.3


0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

2

4

6

8

p

Tact=30, Μ=8.95

Tact=25, Μ=5

Tact=20, Μ=2.65

200 400 600 800 1000x

1

2

3

4

5

p

Tact=30, Μ=60.5

Tact=25, Μ=26.9

Tact=20, Μ=11.3


0 < M+2 (up) and b. M2

0 > M+2 (down)




200 400 600 800 1000x

0.1

0.2

0.3

0.4

JHxL

Tact=30, Μ=8.95

Tact=25, Μ=5

Tact=20, Μ=2.65

200 400 600 800 1000x

0.1

0.2

0.3

0.4

JHxL

Tact=30, Μ=60.5

Tact=25, Μ=26.9

Tact=20, Μ=11.3


0 < M+2 (up) and b. M2

0 > M+2 (down)



Kinetic approach to combustion processes in a recombination …calvino.polito.it/~mmkt/conforto.pdf · Kineticapproachtocombustionprocesses inarecombinationreaction FiammettaConforto

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