lecture 9 - ignition and extinction › sites › cefrc › files › files... · 10!2 100 102 104...

6/26/2011

Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior written permission of the owner, M. Matalon. 1

Lecture 9 Ignition and Extinction

1

x

unburned burned gas uYY

TT

u

uu

0; ;

==

== !!

00/

!

!

YdxdT

S < SL laboratory frame

x

unburned burned gas 0

0/!

!

YdxdTS S uYY

TT

u

uu

==

==

; ;!!

in a frame moving with the wave

Planar flame with volumetric heat loss

Due to heat losses the propagation speed S < SL , and the flame temperatureTf < Ta; both need to be determined.

Heat losses

Heat losses

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d(!u)dx

= 0

!ucpdTdx

!"d 2Tdx2

=Q# ! g(T )

!u dYdx

! !D d 2Ydx2

= !#

x


0/!

!

YdxdTS S uYY

TT

u

uu

==

==

; ;!!

Steady one-dimensional equations

where g(T ) is the volumetric heat loss (J/cm3 s), which is assumed to vanishat T = Tu. For conductive losses, for example, g(T ) = k(T − Tu), while forradiative losses g(T ) = k(T 4 − T 4

u).

3

x


0/!

!

YdxdT

S S uYY

TT

u

uu

==

==

; ;!!

x=0

Using the asymptotic results derived previously, the mathematical formulationbecomes

!

d("u)dx

= 0

"ucpdTdx

# $d2Tdx 2

= #g(T)

"u dYdx

# "Dd2Ydx 2

= 0

on either side of the sheet conditions across the sheet

[[Y ]] = [[T ]] = 0

Q

��ρDdY

dx

��+

��λdT

dx

��= 0

−ρDdY

dx

��x=0−

=λ/cp√ρD

�2ρB/β2 Yu e

−E/2RTf

= ρuSLYu eE/2RTa−E/2RTf

4

SL =λ/cp√ρbD

�2ρB

ρ2uβ2Yu e

−E/2RTa

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Non-dimensionalization

SL (speed), lf =Dth/SL (length), !u,Tu (density, temperature), (" /cp )QYu /# lf2 (heat loss)

s = S/SL, Ta = T̃a/T̃u, Tf = T̃f/T̃u, β = E(T̃a−T̃u)/RT̃ 2a , q = Q/cpT̃u

Boundary conditions

x → −∞ : ρ = T = 1, Y = Yu, u = s

x → +∞ : dT/dx → 0, Y = 0

sdY

dx− d2Y

dx2= 0

ρu = s

Equations

= −β−1(Ta−1)g(T )sdT

dx− d2T

dx2

[[Y ]] = [[T ]] = 0

qLe−1

��dY

dx

��+

��dT

dx

��= 0

where Ta = 1 + qYu

Jump conditions

−Le−1 dY

dx

��x=0−

= Yu exp

�β

2

Ta

Tf

Tf − Ta

Ta − 1

�

5

integrate from −∞ to 0+, namely through and including the reaction zone

Tf = Ta − β−1(Ta−1)ϕ∗ + . . . ,

drop in flametemperature

We have two equations for the determination of s and ϕ∗. We need, however, todetermine first the solution in the preheat and post-flame zones. In particular,in the post-flame zone we need to carry the analysis to O(β−1).

d

dx[s(T + qY )]− d2

dx2

�T + qLe−1Y

�= −(Ta−1)g(T )

−Le−1 dY

dx

��x=0−

= Yue−ϕ∗/2

6

−β−1sϕ∗ −1

Ta−1

dT

dx

��x=0+

= −β−1

� 0+

−∞g(T )dx

s(Tf − Ta)� �� −dT

dx

��x=0+

= −β−1(Ta − 1)

� 0+

−∞g(T )dx

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Y ∼�

Yu

�1− eLe sx

�(x < 0)

0 (x > 0)

T ∼

1 + (Ta − 1)esx + . . . (x < 0)

Ta − β−1

�ϕ∗ +

1

s(Ta−1)g(Ta)x

�

� �� (x > 0)

1

Ta−1

dT

dx

��x=0+

= −β−1

sg(Ta)

reaction zone

preheat zone

O(θ-1)

T

Y adiabatic T = Ta

x

O(β−1)

7

the temperature on the burned side decays to Tu on a much longer scale ~ β

sdT

dx− d2T

dx2= −β−1g(T )

let x = βX

sdT

dX= −g(T )

for g(T ) = k(T − 1), for example,

X = −s

� T

Ta

dT

g(T )

which varies from X = 0 when T = Ta

to X = ∞ when T → 1 (or g(T ) → 0).

T ∼ 1 + (Ta − 1)e−kx/βs

X = −s

� Ta

T

dT

k(T − 1)= − s

kln

Ta − 1

T − 1

reaction zone

preheat zone

O(β-1)

O(β)

T

Y

X = βxx

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sϕ∗ =

� 0

−∞g(T )dx+

1

sg(Ta)

=1

s

� Ta

1g(z)dz + g(Ta)

� ��

total heat loss from thepreheat and post-flamezones, respectively

=1

sqL

−Le−1 dY

dx

��x=0−

= Yue−ϕ∗/2 ϕ∗ = − ln s2

s2 ln s2 = −qL

where

for conductive losses, g(T ) = k(T − 1), the total loss qL = 2k

qL =

� Ta

1g(z)dz + g(Ta)

9

−β−1sϕ∗ −1

Ta−1

dT

dx

��x=0+

= −β−1

� 0+

−∞g(T )dx

(qL)cr = 1/e ≈ 0.37

At extinction the flame speed is nearly 60% of the adiabatic flame speed

s

sext

qL (qL)cr

s2 ln s2 = −qL

sext = e -1/2 ≈ 0.61

unstable

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1.  Safety lamp devised in 1815 by Humphry Davy (Davy lamp) and used in coal mines. The flame in a lamp is surrounded by metal gauze [used to distribute the heat over a large area and ensure that qL > (qL)cr ]

and will not propagate through, thus preventing explosion.

2.  Flame propagation is not possible in narrow tubes, or when r < rcr total conductive losses (through the walls) ~ (2π r)L qL = heat losses per unit volume ~ 2πrL/πr2L → qL ~ 1/r

rcr defines what is known as the “quenching distance”

11

Two-dimensional flames in a channel with conductive losses at the walls

For simplicity, we use here a constant-density model

)(,0 uTTkyT

yY

!=""

=""

Two important parameters: δ = 2a/lf channel’s width (in units of lf ) k a heat loss parameter (k = 0 adiabatic and k → ∞ for cold wall conditions)

burned

Unburned

2a U u

u

TTYY

=

=0

0

=!

!

=!

!

xTxY

x

y

Results of numerical calculations based on a constant density model with finite rate chemistry (β =10) and unity Lewis number are presented below.

12

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reaction rate and temperature contours

δ = 20 δ = 13

the flame survives all values of k flame extinguishes when losses exceed kcr

13

10!2

100

102

104

0.4

0.6

0.8

1

1.2

!=1 2 4

13

15.4

20

40

heat loss

U/SL

  Total extinction in narrow channels of a width smaller than approximately 15lf

  The quenching diameter dq ~ 15lf is in accord with experimental observations   Only partial extinction occurs in wider channels (δ > δ*); the flame persists for all values of k but is confined to the center of the channel.   Calculations how that the dead space ~ 6lF, in agreement with general experimental observations.

Propagation speed as a function of heat loss

14

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Ignition

15

Minimum ignition energy

If energy is deposited locally in a homogeneous combustible mixture, in the formof heat (and/or radicals), ignition will occur only if the energy exceeds a certainminimum value Emin - the minimum ignition energy.

At subcritical conditions the energy diffuses sufficiently fast, faster than the rateat which heat is generated by chemical reaction.

Ignition will occur if the energy added raises the temperature of the gas kernelto the adiabatic flame temperature Ta, in a region comparable in size to theflame thickness lf , so that the energy released is sufficiently fast to overcomethe loss by conduction.

Emin ∼ ρul3f · cp(Ta − Tu)

16

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Calcote et al., Ind. Eng. Chem. 1952

The size of the initial kernel has beenoften related to the quenching distance dq,defined as the minimum distance betweentwo plates that a flame cannot pass through(for example, due to excessive heat loss).This distance scales with thelaminar flame thickness,suggesting a relation

Emin ∼ d3q

which has been verified experimentally.

17

Since dq ∼ lf = λ/ρucpSL,

Emin ∼ S−3L

⇒

Ignition 401

is referred to the literature [17] . However, many of the thermal concepts dis-cussed for homogeneous gas-phase ignition will be fruitful in understanding the phenomena that control dust ignition and explosions.

2 . Ignition by Adiabatic Compression and Shock Waves

Ignition by sparks occurs in a very local region and spreads by fl ame char-acteristics throughout the combustible system. If an exoergic system at stand-ard conditions is adiabatically compressed to a higher pressure and hence to a higher temperature, the gas-phase system will explode. There is little likeli-hood that a fl ame will propagate in this situation. Similarly, a shock wave can propagate through the same type of mixture, rapidly compressing and heating the mixture to an explosive condition. As discussed in Chapter 5, a detonation will develop under such conditions only if the test section is suffi ciently long.

Ignition by compression is similar to the conditions that generate knock in a spark-ignited automotive engine. Thus it would indeed appear that compres-sion ignition and knock are chain-initiated explosions. Many have established

Methane Ethane

Hexane

Butane

Propane

Hexane

Cyclohexane

Cyclopropane

ButanePropane

Diethyl ether

Benzene

Heptane

1 At

1 At

Fraction of stoichiometric percentage of combustible in air0 0.5

0.1

0.2

0.40.50.60.8

1

2

34

0.3

0.20 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

0.40.50.60.8

1

2

34

0.3

1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Min

imum

ene

rgy

(mJ)

FIGURE 7.6 Minimum ignition energy of fuel–air mixtures as a function of stoichiometry (from Blanc et al . [14] ) .

CH07-P088573.indd 401CH07-P088573.indd 401 7/24/2008 5:37:35 PM7/24/2008 5:37:35 PM

more reactive mixturesrequire a smaller ignition energy. fraction of stoichiometric percentage of combustible in air

minim

um

energy

(mJ)

Lewis & von Elbe 1986

We can also deduce a dependence of the minimum ignition energy on pressure,using the relations SL ∼ pn/2−1 and lf ∼ p−n/2 derived previously, with n thereaction order. We find that

Emin ∼ p1−3n/2

While reduction in flame thickness facilitates ignition, the increase in the mix-ture density makes it harder.

Note that the phenomenological relation for Emin may depend on chemical ki-netic considerations (reaction mechanism, surface reactions, etc...) and on thesource and geometry of the ignition kernel (spark ignition, electrical wire, etc...,cylindrical or spherical kernel, etc...) 18

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Thermal Explosion

Thermal explosion and ignition theories are based on the same concept. Theyboth depend on the rate of energy release compared to the rate of energy dissi-pation, but in one case the objective may be to prevent explosion, while in theother to enable ignition.

Typical questions in thermal explosion theory:

• criteria that predict whether explosion occurs, or not

• delay time, or thermal induction period before the abrupt transition to aburning state

• time history of the explosive event

The concept is also similar to chain explosions, depending on the rates of chainbranching and chain termination.

19

Semenov theory1. Adiabatic explosion

Consider a homogeneous combustible mixture at a sufficiently low temperaturein a thermally insulated vessel of volume V.

ρcvdT

dt= BQρY e−E/RT

ρdY

dt= −BρY e−E/RT

T (0) = T0, Y (0) = Y0

d

dt(cvT +QY ) = 0

the adiabatic temperature at constant volume Te is largerthan the adiabatic flame temperature Ta (at constant pressure),because cp > cv.

Te = T0 +QY0/cv

⇒ Y =Te − T

Q/cv

20

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Introduce dimensionless variables, using the initial temperature T0 and thechemical reaction time (Be−E/RT0)−1, the problem reduces to

dT

dt= (Te − T )eβ0(T−1)/T

T (0) = 1

Te = 1 + q, q = QY0/cvT0, β0 = E/RT0.

t = e−β0

�Ei(β0

T )−Ei(β0)−eβ0/Te

�Ei(β0(Te−T )

TTe)−Ei(β0(T−1)

Te)��

Ei(z) =

� z

−∞

ex

xdx

is the exponential integral function (the principle value is needed for z > 0).

The exact solution

The implicit form of the solution, and the use of “ special functions” makes itdifficult to interpret the result.

21

T

1

ttign

β � 1

β � 1

Te

Of primary importance is the case β0 � 1, which yields a sharp transition from

the unburned state to equilibrium.

Ignition is an asymptotic concept

22

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For β0 � 1 the temperature evolves on the “slow time” ∼ β0t , which suggeststhat the “real” characteristic time for nondimensionalization should have been(Bβ0e−β0)

−1. This rescaling yields

dT

dt= β−1

0 (1 + q − T )eβ0(T−1)/T

T = 1 + β−10 ϕ(t) + . . .

dϕ

dt= qeϕ ϕ(0) = 0

ϕ = − ln (1− qt)

tign =cvT0

QY0Bβ0e−β0, β0 = E/RT0

10

T

t tign

T0

T ∼ 1− β−10 ln (1− qt)

23

The ignition time decreases with increasingQ, Y0 and the reaction rate Bβ0e−β0 .

2. Nonadiabatic explosion

Semenov theory

ρcvdT

dt= BQρY e−E/RT − S

Vk(T − T0)

We now allow for conductive heat losses at the surface S of the vessel.

We assume that during the induction period, there is no significant reactantdepletion and the density of the mixture remains constant.

Using the conduction time tc = cvρV/kS as a unit time,

dT

dt= De−β0/T − (T − 1)

heat releaseparameter

D = δeβ0/β0,δ =

QY

cvT0� �� × cvρV/kS

(Bβ0e−β0)−1

� �� conduction time

reaction time24

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Steady statesintersection of the two curves

e−β0/T = 1D (T − 1)

e−β0/T

TT1 Tc

Dc(T

− 1)/D

exist only when D < Dc

For D1 < Dc, a steady state exists with T = T1.

Here energy is dissipated faster than produced,

and the system will reach the steady state T1 as t → ∞.

For D > Dc, the rate of energy generated is faster than the energy dissipationand explosion occurs at a finite time.

The critical conditions are obtained from the tangency conditions

Tc =12 (β0 ±

�β20 − 4β0)

Dc = (Tc − 1)eβ0/Tc

De−β0/T = T − 1

Dβ0

T 2e−β0/T = 1

⇒

25

e−β0/T

TT1 Tc

Dc(T

− 1)/D

Of interest here is the solution with the lower

value of T and, in particular, when β0 � 1. Then

(ρY )c =k(S/V)RT 2

0

QBEeE/RT0−1

pc =k(S/V)R2T 3

0

QY BWEeE/RT0−1critical pressure

minimum massneeded for explosion

QY

cvT0=

k(S/V)B−1eβ0−1

β0cvρ

at criticality

Tc = 1 + β0−1

Dc = β0−1eβ0−1 or δc = e−1

26

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For β0 � 1

T = 1 + β−10 ϕ(t) + . . .

ϕ(0) = 0

dϕ

dt= δeϕ − ϕ

and explosion occurs for δ > δc = e−1

then, ϕ → ∞ when t → texp

texp =

� ∞

0

dϕ

δeϕ − ϕ

An approximation, near the critical conditions, yields

which must be done numerically.

note that texp → ∞ when δ → e−1 as it should.

texp ≈ π

�2e−1

δ − e−1

Estimate of the induction time

⇒

27

Frank-Kamenetski theory

We allow now for spatial temperature variations inside the vessel, but retain theassumptions of no significant reactant depletion (and constant density).

ρcv∂T

∂t− λ∇2T = BQρY e−E/RT

T��walls

= T0

We nondimensionalize the equation as follows:the characteristic size a of the vessel for the spatial coordinate,the conduction time cvρa2/λ for timethe temperature of the walls T0 for temperature.

∂T

∂t−∇2T =

δ

β0eβ0(1−1/T )

T��walls

= 1

δ =QY

cvT0× cvρa2/λ

(Bβ0e−β0)−1=

conduction time

reaction time28

6/26/2011


Steady states

−∇2T =δ

β0eβ0(1−1/T )

The solution T (x, t) would give the temperature distribution in the vessel as afunction of time. The question we will address here is whether steady statesexist, and under what conditions.

For β0 � 1

T = 1 + β−10 ϕ(t) + . . .

⇒∇2ϕ = −δeϕ

ϕ��walls

= 0

Frank-Kamenetskii equation

Solutions were obtained analytically for a slab and a cylinder, and numericallyfor a sphere. In all cases, it was found that solutions exist only for δ < δc.

The time dependent problem will reach a steady state only when δ < δc; other-wise explosion occurs.

29

QY

cvT0

a2Bβ0e−β0

λ/ρcv=

0.88 slab2.0 cyinder2.0 sphere

critical conditions

1 2 3

2

4

6

8

10

ϕmax

δ

30

lecture 9 - ignition and extinction › sites › cefrc › files › files... · 10!2 100 102 104...

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