Coupled Chemical and Thermal Analysis

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Well-Stirred Reactor -1

School of Aerospace Engineering

Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved.AE/ME 6766 Combustion

Coupled Chemical and Thermal

Analysis: Well-Stirred ReactorJerry Seitzman

Methane Flame

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3

Distance (cm)

M o l e F r a c t i o n

0

500

1000

1500

2000

2500

T e m p e r a t u r e ( K )

CH4

H2O

HCO x 1000

Temperature




• Examine flow reactor where Damix= mix / chem<< 1 – high rate of mixing/stirring

– well-stirred reactor

(WSR )

perfectly-stirred reactor

(PSR )

continuously-stirred reactor (CSR )• Useful for examining

– highly mixed IC engines

– low p, low speed reactors (fast molec. diffusion)

– highly turbulent parts of nonpremixed combustors

– residence time issues and high T chemical kinetics

Well-Stirred Reactor






Longwell Reactor (circa 1950)

• Fuel injected from holes in central spinning sphere

ref: Turns

Inlet

Outlets




Multiple Reactors

• Simplified swirling pulverized coal combustorflame into network of well-stirred and plug flowreactors

ref: Kee, Coltrin and Glarborg






• Normally specify inlet conditions

• Interior properties

Conditions

ini

in

in

Y

T

m

,

ou t i

ou t

ou t

Y

T

m

,

iY T ,

Q

– fast mixing means

internal T , Y i

same as outlet

• Results will depend on residence time

• Can develop for various heat transfer conditions – known reactor T

– adiabatic – Q fixed or Q(t )

• Can also examine steady or unsteady inlet conditions




• Species

Governing Equations

• Mass

ini

in

in

Y

T

m

,

ou t i

ou t

ou t

Y

T

m

,

iY T ,

Q

CS

i

CV

i

CM

i

CV

ii Ad nuY dV Y dt d

dt dmdV W ˆ

iniiniout out ini

i Y mY mmmY dt

dY V

,

Reynolds Transport Theorem

iniiniout

i

ii Y mY m

dt

VY d V W

,

CS CV

Ad nudV dt

d ˆ0

out in

mmdt

V d

(III.11)






Governing Equations

• Species (con’t)

iniiniout out ini

i

ii Y mY mmmY

dt

dY V V W

,

ii

iini

ini W

Y Y V

m

dt

dY

,

iniiin

i

ii Y Y mdt

dY V V W

,

res

inin m

m

m

V

• Residence Time

(III.12)

(III.13)




Governing Equations

• Energy

CS CV

inin Ad nuhedV

dt

d QW ˆ

iniinin Y T m

,

,,

ou t i

ou t

ou t

Y

T

m

,

iY T ,

Q

dt

dpV hhm

dt

dhV Q

ininin

dt

dphh

V

m

V

Q

dt

dhin

inin

1

ininin hmhm

dt

hV d Q

dt

dV p

p

inininin

hmhmdt dV p

dt pV d mmh

dt dhV Q






Governing Equations

• Energy (con’t)

– use state eqn’s. for hT

i

i

imix pdt

dY h

dt

dT c

dt

dh

i

iiiniiniin hY hY hh

,,

dt

dpqW h

hhY V

m

dt

dT

c ini

iii

iiiniini

in

mix p

1,,

i

ii

iini

in

i

W Y Y

V

mh

,

chemq

res 1

(III.14)

dt

dphh

V

m

V

Q

dt

dhin

inin

1




Summary: Governing Equations

• Mass

• Species

• Energy

• Variables

– inputs

– unknowns

dt dp

qW h

hhY V

m

dt

dT c in

N

iiii

iiiniini

in

mix p

1

,,

ou t in

mmdt

V d

ii

iini

ini W

Y Y V

m

dt

dY

,

inininiin qT Y m ,,,

,

pT Y N mV iout

,,,,, T R p

mix

– 2+ N ODE’s, 1 algebraic need 2 more constraintse.g., V (t ) and p(t )

(III.11)

(III.12)

(III.14)






Steady-State Solutions

• Mass

• Species

• Energy

• Variables

– inputs

– unknowns

in

N

iiii

iiiniini

in q

W hhhY

V

m

1

,,0

mmmout in 0

ii

iini

W Y Y

V

m

,0

pV qT Y mininini

,,,,,,

T Y N i ,, T R p

mix

– 2+ N coupled nonlinear algebraic eqs. solvable




Steady WSR Analytic Soln

• Species

• Energy

iniiii

resi

iniiini

qW h

hhY

,,

ii

resinii

W Y Y

,

in pi

iniiini T T chhY ,,

P O F

RT E

F

F

F

aeW

Y B

A[Ox ] [F ]

• Simplify

– assume c p constant andsame for all species

– single-step reaction

– Arrhenius rate expression

lean mixture, oxidizer

concentration ~constant






Energy: Steady WSR

• Energy

in

loss

P P OO F F

RT

E

F

F

resin pinOin F m

QW hW hW h Be

W

Y T T cY Y

a

,,

in

loss

P f O f F f

RT E

F

res F

in pm

Qhhh Be

W

Y T T c a

,,,

in

loss RT E

res F in p

m

Q HV BeY T T c a

in

N

iiii

iiiniini

in q

W hhhY

V

m

1

,,0

pin

loss RT E

p

res F

incm

Q HV e

c

BY T T a

heating

value




Normalized Adiabatic Equations

• Species

• Energy

RT E

F resin F F

ae BY Y Y ,

HV ec

BY T T

RT E

p

F res

in

a

T E aea DY Y 1

T E aea DY V H T 1

resinin F F Ba DT T T Y Y Y ;;

,

in pin F inaa T c HVY V H RT E E ,;

T E aea DV H T T 110

• Normalizations

– norm. eq’s.

– combineT E

T E

a

a

ea D

ea DV H

1

1






Solution Limits: Da

• Examine solution limits with Damköhler number

– Da0

slow chemistry

– Da

fast chemistry

T E aea DV H T T 110

)1and(1 Y T

)0and(1 Y V H T

• What happens in between? – examine solutions for various E a

inin F F T T T Y Y Y ;

,




Residence Time Effects

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.01 0.1 1 10 100 1000

Da'

T '

Ea'=2 5 10

Q

HV =4

I

res

inin m

m

m

V

res Ba D

inT T T






1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.01 0.1 1 10 100 1000 10000 100000

Da'

T '

Ea'=2 5 10 15

Residence Time Effects

HV =4

res Ba D

inT T T




Ignition/Extinction

• For low activation energies ( E a /T )

– no distinct ignition/extinction phenomena

– small changes in T do not change reaction rates

• For high activation energy ( E a /T )

– multivalued solutions

– upper (>Q) and lower (< I ) branches stable

– middle branch unstable (hysteresis);

leads to extinction and ignition temperatures/

residence times

– for large E a/T , Ignition/Extinction separate more






Blowout Limits

• Can use this approach to model minimum residence

times (maximum mass flow rates) to prevent

blowout in gas turbine combustors

Stable1

nVpm

0.4

Blowout




Transient Reactor: Oscillating Ignition

• Sinusoidal forcing

of reactor volume

close to ignition

conditions

• Low frequencies

shorter delays

%4V V rms

H2/“Air”; =1; 1atm

ref: Kee, Coltrin and Glarborg






Transient Reactor: Oscillating Ignition

• Net radical production increase during each cycle

– nonlinear T dependence produces more radicals

during compression than lost during expansion

ref: Kee, Coltrin and GlarborgH2/“Air”; =1; 1atm

2500 Hz

Coupled Chemical and Thermal Analysis

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