Well Stirred Reactor
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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
Well-Stirred Reactor -2
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
• 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
Well-Stirred Reactor -3
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
Longwell Reactor (circa 1950) • Fuel injected from holes in central spinning sphere
ref: Turns
Inlet
Outlets
Well-Stirred Reactor -4
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
Multiple Reactors
• Simplified swirling pulverized coal combustor flame into network of well-stirred and plug flow reactors
ref: Kee, Coltrin and Glarborg
3
Well-Stirred Reactor -5
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
• 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
Well-Stirred Reactor -6
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
• 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
Well-Stirred Reactor -7
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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)
Well-Stirred Reactor -8
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -9
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -10
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -11
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -12
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -13
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -14
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -15
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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 ;,
Well-Stirred Reactor -16
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -17
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -18
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -19
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -20
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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
Well-Stirred Reactor -21
School of Aerospace Engineering
Copyright © 2004-2005 by Jerry M. Seitzman. All rights reserved. AE/ME 6766 Combustion
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