reacting systems new
TRANSCRIPT
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Chemical Kinetics
1
Global vs. Elementary Reactions
Law of Mass Action
Arrhenius Law
Relation between forward and reverse reaction rates
Steady State approximation
from S.R Turns
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Hydrocarbon Combustion (Turns p. 157)
2
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H2-O2 Combustion (Turns p. 117)
3
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Reactor models
Constant pressure reactor
T = T(t)
[Xi] = [Xi](t)
V = V(t)
Constant volume reactor
T = T(t)
[Xi] = [Xi](t)
P = P(t)
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Reactor models (contd.)
Well-stirred reactor
T = constant[Xi] = constant
P = constant
Plug-flow reactor
T = T(x)[Xi] = [Xi](x)
P = P(x)
V = V(x)
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6
Assumptions:
No temperature gradients
No composition gradientsi.e. T and [Xi] are functions of time
Known:
0
0
)0(
)0(
ii XtX
TtT
Constant Pressure Reactor
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Conservation of energy (First Law of Thermodynamics):
dt
dvP
dt
dhm
dt
dvmP
dt
dumWQ
dt
dh
m
Q
Constant Pressure Reactor
Expressing system chemical enthalpy in terms of chemical composition,
m
hN
m
Hh i
ii
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i i
ii
ii
dt
hdN
dt
dNh
mdt
dh 1
Assuming Ideal Gas behavior (h is a function of T only) :
dt
dT
cdt
dT
T
h
dt
hdip
ii
,
Constant Pressure Reactor (contd.)
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N
j
i
M
k u
akkkikii
kjk X
TR
ETA
11
'
,
"
,
,'
exp
The rate of change ofNi : [ ]i
i i i
dNN V X Vdt
Where,
Substituting these expressions in the First Law,
i
pi
i
ii
icX
hVQ
dt
dT
Constant Pressure Reactor (contd.)
1
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Where,
i
ii MWX
mV
T
T
ipifi
ref
dTchh ,0,
Constant Pressure Reactor (contd.)
dt
dV
VXdt
dV
VNdt
dN
Vdt
V
Nd
dt
Xdiii
i
i
i 1112
The rate of change of[Xi] is given by :
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dt
dT
TX
X
dt
Xd
ii
i
i
iii 1
By using the ideal gas law :
dt
dT
Tdt
dN
Ndt
dV
V i
i
i
i
111
Constant Pressure Reactor (contd.)
2
i ui
PV N R T Differentiating
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Constant Pressure Reactor (contd.)
Solution Methodology:
System ofFirst order ODEs
Integration routine capable of handling stiff equations
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Assumptions:
No temperature gradients
No composition gradients
i.e. T and [Xi] are functions of time
Known:
0
0
)0(
)0(
ii XtX
TtT
Constant Volume Reactor
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Conservation of energy (First Law of Thermodynamics):
dt
dumWQ
dt
du
m
Q
Constant Volume Reactor
Expressing system chemical internal energy in terms of chemical
composition,
i
upi
i
ii
i
iu
RcX
hTRVQ
dt
dT
i
1
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Constant Volume Reactor (contd.)
Also, the rate of change of pressure (using Ideal Gas Law):
The rate of change of[Xi] is given by :
ii
dt
Xd
iiu
iiu dt
dT
XRTRdt
dP
2
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Well-stirred reactor
Conservation of species for the integral CV :
Assumptions: Steady state operation
Steady flow operation
System isperfectly mixedand
homogenous in compositionini
ini
h
Y
m
,
,
outi
outi
h
Y
m
,
,
outiinii
cvi mmVmdt
dm,,
"',
Rate of
accumulation
of mass i inCV
Rate of
generation of
mass i in CV
Mass
flow ofi
into CV
Mass flow
ofi out of
CV
Known: iniYVm ,,,
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Well-stirred reactor (contd.)
Since diffusional mass flow rate is negligible,
0)( ,,"' outiinii YYmVm
N
j
i
M
k u
akkkikiii
kjk XTR
ETBMWm11
'
,
"
,
,'
exp
Where,
ii Ymm Hence, conservation of mass :
fori = 1,2,,N
Since the reactor is homogeneous, the mass fraction at the outlet is
equivalent to that inside the reactor
TXfTXfVmouticvii
,,"'
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Well-stirred reactor (contd.)
Conservation of Species providesNequations withN+1 unknowns
Additional equation from conservation of energy
inout hhmQ
N
j
jj
ii
i
MWX
MWX
Y
1
Where, [Xi]and Y
iare related as
Conservation of energy for steady state,
steady flow conditions:
In terms of individual species :
N
i
iniini
N
iiouti
ThYThYmQ1
,
1
,
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Well-stirred reactor (contd.)
T
T
ipifi
ref
dTchTh ,0,Where,
Conservation of mass and conservation of energy are simultaneously
solved forTand Yi,out
Solution Methodology:
Coupled non-linear algebraic equations, rather than system ofODEs
Mass generation rate depends only on Yior[X]i
Generalized Newtons methodused to solve the system
"'im
Mean residence time, tR
mVtR /
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Plug flow reactor
Assumptions:
Steady state and steady flow operation
No mixing in the axial direction
Uniform properties in direction perpendicular to flow
Ideal frictionless flow
Ideal gas behavior
X
x
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Plug flow reactor (contd.)
Conservation of mass :
Known quantities :
", ( ), ( ) (nozzle or diffuser), ( )i
m k T A x Q x
0
dx
uAd
Conservation of momentum: 0dx
duu
dx
dP
Conservation of energy:
2
"
2 0m
ud h
Q Pdx m
(Pm = perimeter)
Conservation of species: "'
ii m
dx
uYd
fori = 1,2,,N
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Plug flow reactor (contd.)
Ideal gas equation of state:
1
1
N
i i
imix
mix
u
MW
YMW
MW
TR
P
Where,
Hence,N+4 unknowns (Yis,, P, T, u)
withN+4 equations
Usable forms of equations
0111
dx
dA
Adx
du
udx
d
From conservation of mass
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Plug flow reactor (contd.)
0 dx
duudx
dP
0"
m
PQ
dx
duu
dx
dh
dx
dYh
dx
dTc
dx
dh iN
i
ip
1
From ideal gas calorific
equation of state h=h(T, Yi)
01111 dx
dMWMWdx
dTTdx
ddxdP
Pmix
mix
From conservation of momentum
From conservation of energy
From ideal gas law
dx
dY
MW
MW
dx
dMW iN
i i
mixmix
1
2 1 FromMWmix
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Combustion system modeling
Conceptual drawing of a Gas turbine combustor
Fuel
Air
ProductsWSR1 WSR2 PFR