Well Stirred Reactor

1

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 Reactor

Jerry Seitzman

Methane Flame

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3

Distance (cm)

Mo

le F

racti

on

0

500

1000

1500

2000

2500

Te

mp

era

ture

(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

2




Longwell Reactor (circa 1950) • Fuel injected from holes in central spinning sphere

ref: Turns

Inlet

Outlets




Multiple Reactors

• Simplified swirling pulverized coal combustor flame into network of well-stirred and plug flow reactors

ref: Kee, Coltrin and Glarborg

3




• Normally specify inlet conditions

• Interior properties

Conditions

ini

in

in

Y

T

m

,

outi

out

out

Y

T

m

,

iYT ,

Q

– fast mixing means

internal T, Yi

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

,

outi

out

out

Y

T

m

,

iYT ,

Q

CS

i

CV

i

CM

i

CV

iiAdnuYdVY

dt

d

dt

dmdVW ˆ

iniinioutoutini

i YmYmmmYdt

dYV

,

Reynolds Transport Theorem

iniiniout

i

iiYmYm

dt

VYdVW

,

CSCV

AdnudVdt

dˆ0

outin

mmdt

Vd

(III.11)

4




Governing Equations

• Species (con’t)

iniinioutoutini

i

iiYmYmmmY

dt

dYVVW

,

ii

iini

iniW

YYV

m

dt

dY

,

iniiin

i

iiYYm

dt

dYVVW

,

res

ininm

m

m

V

• Residence Time

(III.12)

(III.13)




Governing Equations

• Energy

CSCV

ininAdnuhedV

dt

dQW ˆ

iniininYTm

,,,

outi

out

out

Y

T

m

,

iYT ,

Q

dt

dpVhhm

dt

dhVQ

ininin

dt

dphh

V

m

V

Q

dt

dhin

inin

1

inininhmhm

dt

hVdQ

dt

dVp

p

inininin

hmhmdt

dVp

dt

pVdmmh

dt

dhVQ

5




Governing Equations

• Energy (con’t)

– use state eqn’s. for hT

i

i

imixpdt

dYh

dt

dTc

dt

dh

i

iiiniiniinhYhYhh

,,

dt

dpqWh

hhYV

m

dt

dTc ini

iii

iiiniini

in

mixp

1,,

i

ii

iini

in

i

WYY

V

mh

,

chemq

res1

(III.14)

dt

dphh

V

m

V

Q

dt

dhin

inin

1




Summary: Governing Equations

• Mass

• Species

• Energy

• Variables

– inputs

– unknowns

dtdp

qWh

hhYV

m

dt

dTc in

N

iiii

iiiniini

in

mixp

1

,,

outin

mmdt

Vd

ii

iini

iniW

YYV

m

dt

dY

,

inininiinqTYm ,,,

,

pTYNmViout

,,,,, TRp

mix

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

(III.11)

(III.12)

(III.14)

6




Steady-State Solutions

• Mass

• Species

• Energy

• Variables

– inputs

– unknowns

in

N

iiii

iiiniini

inq

Wh

hhYV

m

1

,,0

mmmoutin

0

ii

iini

WYY

V

m

,0

pVqTYmininini

,,,,,,

TYNi,,

TRpmix

– 2+N coupled nonlinear algebraic eqs. solvable




Steady WSR Analytic Soln

• Species

• Energy

ini

iii

resi

iniiini

qWh

hhY

,,

ii

resinii

WYY

,

inp

iiniiini

TTchhY ,,

POF

RTE

F

F

F

aeW

YB

A[Ox] [F]

• Simplify

– assume cp constant and

same for all species

– single-step reaction

– Arrhenius rate expression

lean mixture, oxidizer

concentration ~constant

7




Energy: Steady WSR

• Energy

in

loss

PPOOFF

RT

E

F

F

resinpinOinFm

QWhWhWhBe

W

YTTcYY

a

,,

in

loss

PfOfFf

RTE

F

resF

inpm

QhhhBe

W

YTTc a

,,,

in

lossRTE

resFinpm

QHVBeYTTc a

in

N

iiii

iiiniini

inq

Wh

hhYV

m

1

,,0

pin

lossRTE

p

resF

incm

QHVe

c

BYTT a

heating

value




Normalized Adiabatic Equations

• Species

• Energy

RTE

FresinFF

aeBYYY

,

HVec

BYTT

RTE

p

Fres

in

a

TEaeaDYY 1

TEaeaDYVHT 1

resininFFBaDTTTYYY ;;

,

inpinFinaaTcHVYVHRTEE

,;

TEaeaDVHTT 110

• Normalizations

– norm. eq’s.

– combine TE

TE

a

a

eaD

eaDVH

1

1

8




Solution Limits: Da

• Examine solution limits with Damköhler number

– Da0

slow chemistry

– Da

fast chemistry

TEaeaDVHTT 110

)1and(1 YT

)0and(1 YVHT

• What happens in between?

– examine solutions for various Ea

ininFF TTTYYY ;,




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

ininm

m

m

V

resBaD

inTTT

9




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

resBaD

inTTT




Ignition/Extinction

• For low activation energies (Ea /T)

– no distinct ignition/extinction phenomena

– small changes in T do not change reaction rates

• For high activation energy (Ea /T)

– multivalued solutions

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

– middle branch unstable (hysteresis);

leads to extinction and ignition temperatures/

residence times

– for large Ea/T, Ignition/Extinction separate more

10




Blowout Limits

• Can use this approach to model minimum residence

times (maximum mass flow rates) to prevent

blowout in gas turbine combustors

Stable 1

nVpm

0.4

Blowout




Transient Reactor: Oscillating Ignition

• Sinusoidal forcing

of reactor volume

close to ignition

conditions

• Low frequencies

shorter delays

%4VVrms

H2/“Air”;=1; 1atm

ref: Kee, Coltrin and Glarborg

11




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 Glarborg H2/“Air”;=1; 1atm

2500 Hz

Well Stirred Reactor

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