Transcript RADIATION AND COMBUSTION PHENOMENA

COMBUSTION



PROF. SEUNG WOOK BAEK
 DEPARTMENT OF AEROSPACE ENGINEERING, KAIST, IN KOREA
ROOM: Building N7-2 #3304
TELEPHONE : 3714

Cellphone : 010 – 5302 - 5934
[email protected]
http://procom.kaist.ac.kr

TA : Bonchan Gu

ROOM: Building N7-2 #3315
TELEPHONE : 3754
Cellphone : 010 – 3823 - 7775
[email protected]



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
CONSERVATION EQUATIONS OF
MULTICOMPONENT GASES
CONSERVATION OF
MASS
MOMENTUM
ENERGY
OF A GASEOUS MIXTURE
OF DIFFERENT CHEMICAL SPECIES
NEW PHENOMENA
(MASS) DIFFUSION
TRANSPORT OF ENERGY BY DIFFUSION
CREATION OR DESTRUCTION OF CHEMICAL SPECIES
BY REACTION
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
HOMEWORK : (pg. 390) #3, #18, #24, and #27
Attached a slide for your information
DEFINITION OF VARIOUS VELOCITIES
GAS CONSISTS OF MANY RANDOMLY MOVING MOLECULES
v
v
= VELOCITY OF PARTICLE OF STH KIND
o = MASS AVERAGE VELOCITY OF MIXTURE,
VELOCITY OF THE C.G OF A FLUID ELEMENT
V s = PECULIAR VELOCITY OF PARTICLE OF STH KIND,
VELOCITY OF THE PARTICLES W.R.T
o
s
Vs  v s  v o
v
DEAL WITH DISTRIBUTION FUNCTION FOR PARTICLES
f S (r, vs , t ) = PARTICLE DISTRIBUTION FUNCTION
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
f dv
s
s
represents how many particles per unit volume
in the range of dv s   v s  dv s
 PROBABLE NUMBER OF PARTICLES OF KIND S PER
UNIT VOLUME, WHICH AT TIME t HAVE A VELOCITY
IN UNIT RANGE ABOUT v s
ns  NUMBER DENSITY, PARTICLES OF STH KIND PER
UNIT VOLUME
vs  AVERAGE VELOCITY OF PARTICLES OF STH KIND
f
s
v (r , t ) 
s
1
v f

n
s
s
n (r , t )   f
(r , vs , t )d v s
s
s
(r , vs , t )d v s
s

1
(i v

n
sx
 j v sy  k v sz ) f d v sxd v syd v sz
s
MASS AVERAGE VELOCITY
PROPULSION AND COMBUSTION LABORATORY
s
v  v (r , t )
o
o
COMBUSTION ENGINEERING
 vo    s vs
s
WANT TO KNOW AVERAGE MOTION OF A PARTICULAR SPECIES
W.R.T.
v
o
= AVERAGE PECULIAR VELOCITY (DIFFUSION VELOCITY)
THE DIFFUSION VELOCITY OF CHEMICAL SPECIES s IS THE RATE
OF FLOW OF MOLECULES OF s W.R.T THE MASS AVERAGE
VELOCITIY OF THE GAS.
1
(  )
ns  v s v o
Vs
f dv  v  v
s
s
s
o
NET RATE OF MASS TRANSFER BY DIFFUSION IS ZERO, I.E.
 V   v  v 
s
s
s
s
s
s
s
PROPULSION AND COMBUSTION LABORATORY
s
o
v  v  0
o
o
COMBUSTION ENGINEERING
CONSERVATION OF MASS
n s
t
 K S    (n sv s ) 
n s
t
   (n sv s )  K S
LET K  NO. OF PARTICLES OF THE STH KIND FOUND BY
CHEMICAL REACTION PER UNIT VOLUME, PER UNIT TIME
S
LET ms = MASS OF ONE PARTICLE OF SPECIES, s
MULTIPLY BY
m
s
 s
    s (vo  Vs )  ms K S  ws
t
w
s
MASS OF STH SPECIES FORMED PER UNIT VOLUME PER
UNIT TIME DUE TO CHEMICAL REACTION
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
 s
    s (vo  Vs )  ms K S  ws
t
w  0
s
 s
 [ t     s (vo V s)  ws]  0
   sV s    ( sV s)  0
s
s
MASS IS CONSERVED IN CHEMICAL REACTION
s


   (  vo ) 
 vo     vo  0
t
t
D
   vo  0
Dt

y

s


s
OVERALL MASS CONSERVATION
D 
  vo 
Dt t
 
s
y
s
Dys
y
  ( s  vo ys )    (  sV s )  w s    (  ysV s )  w s
Dt
t
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
DIFFUSION VELOCITY
V
s
DEPENDS ON f s .
Maxwell-Boltzmann DISTRIBUTION FUNCTION
(GAS IN EQUILIBRIUM)
1
V
s

n V
f d vs  0
s s
AS A PERTURBATION SOLUTION ABOUT Boltzmann DISTRIBUTION
Chapmann & Enskog OBTAINED
2
Vs
n mD d
n  
s
p ps
p
sp
p

1
nm
s
D  ln T
T
s
s
T
D = MULTI COMPONENT THERMAL DIFFUSION COEFFICIENT
s
V
s
= DIFFUSION VELOCITY IN MULTI COMPONENT MIXTURE
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
n
n
d  ( )  ( 
p
p
n
p
n
nm
p

p
) ln P 
nm
p
p
p
( X
1
p
 n X )
l
m
l
l
p
X = EXTERNAL FORCE ACTING ON MOLECULE l
l
nP
 XP 
n
MOLE FRACTION OF SPECIES P
mP  nP
 yP 

PPV  N P RT
MASS FRACTION OF SPECIES P
NP
 CP 
V
CONCENTRATION OF SPECIES P

PP
X P
 P
RT
RT
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
DIFFUSION FLUX ARISES FROM
(i)
(ii)
(iii)
(iv)
Gradient of concentration
Pressure gradient
Flux due to external forces such as electromagnetic
forces or gravity
Temperature gradient : Thermal diffusion
IN MANY PROBLEMS, THERMAL & PRESSURE DIFFUSIONS
ARE NEGLECTED.
nm
EXTERNAL FORCE DIFFUSION = 
P
p
PROPULSION AND COMBUSTION LABORATORY
p


1
X
  n X 

m


p
l
l
l
p
COMBUSTION ENGINEERING
FOR GRAVITY ONLY
X m g
p
p
g = ACCELERATION OF GRAVITY
m g
1
X
 n X 
 nm g  g  g  0
m
m
p
p
l
l
l
p
l
l
l
p
BINARY DIFFUSION
CONSIDER INTERDIFFUSION BETWEEN SPECIES 1 AND 2
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
2
Vs
n mD d
n  
s
n
p ps
n
d  ( )  ( 
p
p
n
p
sp
p
nm
n
p

p
p

1
D  ln T
T
nm
s
) ln P 
s
s
nm
p
( X
p
p
1
 n X )
l
p
m
l
l
p
 n n nm 

n
nm 
1
  X
V
m D    
 n X  n X 
 ln P 
n
 
P 
m

 n n
n n (m  m )
1
T

D1  ln T
n
n1m1

2
2
1
2
2
2
2
2
2
12
2
1
1
1
2
2

2
1
2
1
2
IF m1  m2 , PRESSURE DIFFUSION OF SPECIES 1 OCCURS IN THE
DIRECTION OF INCREASING PRESSURE IN SPECIES 2
IF m1  m2 , THERE IS NO PRESSURE DIFFUSION.
n
V 
mD
n
2
2
1
2

21
 n n nm 
nm




ln
P





 
P
 n n
1
1
1
1
1
1


1
  X
 n X  n X 
m



1
1
1
2
2

1
1
T
D2  ln T
n2 m2
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
 n n nm 
n
nm
V
m D    
 ln P 
n
n
n

P



1
T

D1  ln T
n1m1
2
2
1
2
2
2
2
12
1
2
2


1
  X
 n X  n X 
m



2
1
1
2
2

2
IN THE ABSENCE OF P, T , WHILE NEGLECTING X
2
2
n2
n2
 n2 
 n1  V  n m D   n1    n m D   n2 
V1 
m2 D12    
m2 D12  
2
1 21 
1 21 


n

n
n

n




n1 
n
n

n
2
2
 
 
1
p
n1  n2  n
n1 n2
 1
n n
n 
n 
 1    2 
n
n
n1m1V1  n2 m2V2  1V1  2V2  0

n 2 m1m2

2
n 2 m1m2  n2 
 n2  n m1m2
 n2 
D12  
D21   D12  D21 
   0


n
n
n
 D12  D21
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
2
n2
n2
 n2  n
V1 
m2 D12   
m2 D12X 2  
m2 D12X 1
n1
n1
 n  n1
LET
k

T
T
D1
(THERMAL DIFFUSION RATIO) = 2
n m1m2 D12


n
V 
m D d  k  ln T
n
2
1
2
12
2
T
1
MASS FRACTION
mS nS
ns
 yS  MASS FRACTION
 X s  MOLE FRACTION,
 mS nS
n
s
1V1  
2
n m1 m2

D12X 1
1V1  D12y1
V1   D12 ln y1
X1 
y1m2
m1  y1 (m2  m1 )
 /W
 /m

 /W   /W  / m   / m
Fick’s LAW (EXACT FOR BINARY DIFFUSION)
PROPULSION AND COMBUSTION LABORATORY
X 
1
1
1
1
1
1
1
2
2
1
1
2
2
COMBUSTION ENGINEERING
APPROXIMATION: FREQUENTLY USED
(AS A DESPERATE MEASURE)
s Vs  Ds ys
PROPULSION AND COMBUSTION LABORATORY
Vs   Ds  ln ys
COMBUSTION ENGINEERING
CONSERVATION OF MOMENTUM
VERY FEW CHANGES DUE TO CHEMICAL REACTIONS
ui
ui
1  ij
uj
 Xi 
t
x j
 x j
ij  ij p   ij   ij R
p
 ij

ij
STRESS TENSOR
= THERMODYNAMIC OR HYDROSTATIC PRESSURE
= KRONECKER DELTA
= VISCOUS STRESS TENSOR
 u u  2
u
     
 

 3

x

x
x


i
j
k
ij
j

i
k
= BULK VISCOSITY ACCOUNTING FOR
INTERNAL RELAXATION
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
ij  ij p   ij   ij
R
 ui u j  2
u
 ij   

        ij k

 x

xk
 j xi  3
 u v w 








3
p

2

 xx yy zz 
  


x

y

z


FOR MONATOMIC GAS,

R
ij
Xi
  = 0 (Stokes HYPOTHESIS)
= RADIATION STRESS TENSOR
= i th COMPONENT OF BODY FORCE
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
(   )  v   : (v )
CONSERVATION OF ENERGY
o
o
D
v
 (e  )  v  p  p  v    (  v )    q
Dt
2
   X  (v  V )  Q
2
0
0

0
0
= VISCOUS STRESS DIADIC
S
S
S
o
S
v
0
q = HEAT FLUX VECTOR
x
Q = RATE OF HEAT GENERATION PER UNIT MASS E.G. : RADIATION

i
ij
j
e = INTERNAL ENERGY PER UNIT MASS
h e
p
REMEMBER FORMATION ENERGY INCLUDED !

D
v02
P
 (h  ) 
   (  vo )    q    s Xs  (vo  VS )   Q
Dt
2
t
 X V  X V  0
IF X IS X FOR ALL SPECIES,
S
s
S
S
S
S
S
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
THE HEAT FLUX VECTOR
q
= ENERGY TRANFERRED PER UNIT AREA PER UNIT TIME DUE
TO TRANSPORT EFFECTED AT A SURFACE
PLANE SURFACE 
T
x DIRECTION
i
qi   
T
   shs (V s )i  Diffusion Thermo Term
xi
s
Dufour effect
Conduction
Boltzmann CONSTANT
Ci
x
  sV shs
i
q    T 
s
Energy Transport by Diffusion Flux
PROPULSION AND COMBUSTION LABORATORY
kT

n
n p DsT
(Vs  V p )

s
p ms Dsp
Dufour effect
COMBUSTION ENGINEERING
THE ADIABATIC REACTION
SECOND ORDER REACTION TO REPRESENT THE ACTUAL
KINETICS
k
A 
B 
 C


Fuel
Oxidant
d ( A)
 k ( A)( B)
dt
Product
k : specific reaction rate constant
k  K T 1/2 exp( E / RT ) typical of Arrhenius type of reaction
WHERE E : ACTIVATION ENERGY
K  : STERIC FACTOR
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
d ( A)
 k ( A)( B)
dt
yA 
( A) 
WA
k  K T 1/ 2 exp(  E / RT )
yB 
( B) 
WB
  W
A
A
d ( A)
dt
MIXTURE OF IDEAL GASES
pW

RT
2
2
y
y
p
W
  A'''   K T exp( E / RT ) A B 2
WB R T 2
1/2
AFTER REARRANGEMENT
A'''   p2 KT 3/2 yA yB exp(E / RT )
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING

 A
   p KT y y exp( E / RT )

2
A
3 / 2
A
B
T
SINGLE STEP KINETIC IS OFTEN GENERALIZED TO THE FORM
   AT yAa yBb exp( E / RT ) p m
'''
A
n
WHERE A,a,b,n,m,E ARE DETERMINED BY COMPARISON WITH
EXPERIMENTS OR DETAILED KINETICS CALCULATION.
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
SIMPLY CHEMICALLY REACTING SYSTEM (SCRS)
FOR A SYSTEM OF UNIT MASS
f  MASS OF FUEL
1  f  MASS OF OXIDANT
f st  STOICHIOMETRIC FUEL MASS
h  HEAT RELEASED BY COMBUSTION OF A UNIT MASS OF
FUEL FOR THE REACTION GIVEN
cT  y h  INITIAL ENERGY
0
FOR THE SAME SPECIFIC HEAT
Ao
FOR AN ADIABATIC REACTION
cT  y h  cT  y h
0
Ao
(T  T ) 
O
A
    h
(y  y )
h 
c
c
Ao
PROPULSION AND COMBUSTION LABORATORY
A
AO
A
COMBUSTION ENGINEERING
cT  y h  cT  y h
0
Ao
A
LET T  CONDITIONS WHEN COMBUSTION IS COMPLETE
y  THIS DEPENDS ON f AND f

y 
y 
T T
h
1
A1
ST
B1
C1
1
1
y
B1

T T
st
T T 
0
st
0
y
A1
0
f  f
st
c
f
st
f
y  0 y  1
f
f f
y 
y 0
1 f
B1
A1
st
f
1
f
f f
st
st
A1
B1
st
st
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
cT  y h  cT  y h
0
f  f
Ao
st
A
(T  T )  f
1
0
h f
 (T  T )
c
f
ST
0
st
f f
st
1 f
(T  T )
1 f
(y  y )
(T  T ) 
h
c
h
T T 
f
c
T T 
1
0
st
0
st
USING
Ao
A
O
st
0
st
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
DIMENSIONLESS TEMPERATURE

T T
T T
1

0  1
0
DURING REACTION
0
: PROGRESS VARIABLE
(y  y )
(T  T ) 
h
c
Ao
A
O
BECOMES
c(T  T )
y y 

h
1
A
0
A0
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
T T 
st
f  f
c(T  T )
y y 

h
1
A
AS
st
y 0
y  1
B1
A1
st
f
f
st
f f
0
A0
 1
0
h
f
c
y 
A1
st
f f
1 f
y 0
st
B1
st
f  f  y 0

f f

f

f

y


1 f

st
A1
st
A1
st
f f
y 
f
y 0
st
c(T  T ) 1  f
y y 
(
)
h
f
f  f

f  f
PREVIOUSLY, WE OBTAINED
   p KT y y exp( E / RT )
1
B
0
st
B0
st
st
st
B1
st
st
B1

3 / 2
2
A
A
A
B
c(T  T )
 ]
h
c(T  T ) 1  f
E
[y 
(
) ] exp[
]
h
f
R(T   (T  T )
   Kp [ (T  T )  T ] [ y 

3 / 2
2
1
0
1
0
0
1
0
Ao
st
Bo
st
PROPULSION AND COMBUSTION LABORATORY
0
1
0
COMBUSTION ENGINEERING
AS 
 1,
EXPONENTIAL TERM INCREASES DRASTICALLY
HOWEVER,
y OR yB  0 , SO THAT
A
  0

A

 A

1.0
PROPULSION AND COMBUSTION LABORATORY
COMBUSTION ENGINEERING
Homework #1
4th reference : M. Kanury (pg.390) #3, #18, #24, and #27
1.
Calculate the pressure of 28g of hydrogen contained in 1 liter vessel at 25℃
2. Hydrogen Peroxide is used sometimes as an oxidizer in special power plants such as
torpedoes and rockets. Determine the standard enthalpy of combustion per gram mass
of the fuel for the following combustion reaction :
H2O2(l) + 0.25CH4(g) → 1.5H2O(g) + 0.25CO2(g)
Also Compare this enthalpy of combustion with that calculated for methane burning in
pure oxygen. Assume h0f 298 = -187.9 kJ/mole of H2O2(l).
3. A mixture at 298oK and 1 atm. Pressure consists of 1 mole of H2 and 0.5 mole of O2.
They are slowly heated to 2,500oK keeping pressure constant. What is the final
equilibrium composition?
4. A heated tube reactor is operated at 2,500 oK. The flowing mixture initially contains 2
moles of H2O and 1 mole each of O2 and N2. If the total pressure is approximately 2 atm.
And the outlet equilibrium mixture contains only H2O, O2, H2, N2 and OH, calculate the
outlet composition.