notes lecture 5 - cefrc.princeton.edu
TRANSCRIPT
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 1
Lecture 5
Stoichiometry Dimensionless Parameters
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NX
i=1
⌫0iMi �NX
i=1
⌫00i Mi
dni
�00i � �0i=
dnj
�00j � �0j
or in terms of the partial masses
dmi
(�00i � �0i)Wi=
dmj
(�00j � �0j)Wj
dYi
(�00i � �0i)Wi=
dYj
(�00j � �0j)Wj
Stoichiometry
Given the reaction
we have seen that there is a relation between the change in the number of moles
of the species; i.e., for any two species i and j
Since the total of mass (unlike the total number of moles) in the system is
unchanged by chemical reaction, we also have
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Consider a global reaction describing the combustion of a single fuel
for example, the combustion of a hydrocarbon fuel CmHn
dYF
�FWF=
dYO
�OWO
⌫F Fuel + ⌫O Oxidizer ! Products
⌫0F CmHn + ⌫0O O2 ! ⌫00CO2CO2 + ⌫00H2O
H2O
with the stoichiometric coe�cients�0F = 1, �0O = m+ n/4, �00CO2
= m, �00H2O= n/2
where �0F was taken equal to one, arbitrarily.
which may be written as (primes are unnecessary)
then �O/�F is the ratio of the stoichiometric coe�cients, and
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dYF
�FWF=
dYO
�OWO
YF � YF u
�FWF=
YO � YOu
�OWO
YOu
YF u
�����st
=�OWO
�FWF⌘ �mass-weighted
stoichiometric ratio
between the initial unburned state (subscript u) and a later state
Integrating
A fuel-air mixture is referred to as a stoichiometric mixture, if the fuel-to-oxygenratio is such that both reactants are entirely consumed; i.e. when combustionto CO2 and H2O is completed.
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In general, the initial state may not be at stoichiometry. A measure of the
departure from stoichiometry is given by the equivalent ratio �
0 < � < 1
for combustion in air, the equivalence ratio is often expressed as the fuel-to-air
ratio
� = 1 stoichiometric mixture
� < 1 lean mixture (in fuel)
� > 1 rich mixture (in fuel)
� =YF u/YOu
YF u/YOu
��st
=YF u/YOu
⌫FWF /⌫OWO=
⌫YF u
YOu
� =F/Air
F/Air��st
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Adiabatic Flame Temperature
If a given combustible mixture is made to approach chemical equilibrium by
means of an isobaric, adiabatic process, then the temperature attained by the
system is the adiabatic flame temperature Ta.
For an adiabatic, isobaric process dh = 0. Integrating from the unburned to the
burned state
hu = hb
NX
i=1
Yiuhi
=NX
i=1
Yibhi
NX
i=1
Yiuho
i
+
ZTu
T
o
cp
udT =
NX
i=1
Yibho
i
+
ZTb
T
o
cp
bdT
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NX
i=1
�Yiu
� Yib
�ho
i
=
ZTb
T
o
cp
bdT �
ZTu
T
o
cp
udT
cpu=
NX
i=1
Yiucpi(T ) , cpb=
NX
i=1
Yibcpi(T )
where the specific heats are those of the mixture, calculated with the mass
fractions of the unburned/burned gas, respectively
dYi
(�00i � �0i)Wi=
dYj
(�00j � �0j)Wj
can be integrated to give
For a one-step global reaction, where we consider j = 1 to be the fuel, denoted
by F, the equation
Yiu � Yib =�YF b
� YF u
� (�00i � �0i)Wi
�FWF
dYi
(⌫00i � ⌫0i)Wi= � dYF
⌫0FWF
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NX
i=1
�Yiu
� Yib
�ho
i
| {z }=
ZTb
T
o
cp
bdT �
ZTu
T
o
cp
udT
YF b
� YF u
�F
WF
NX
i=1
(�00i
� �0i
)Wi
ho
i
| {z }�Q
k
Z Tb
T o
cpbdT �
Z Tu
T o
cpudT =
YF u� YF b
�FWFQ
Z Ta
Tu
cpbdT =
YF u
⌫FWFQ
Using this last relation in the conservation of energy equation:
If the reference temperature T o
= Tu, for complete combustion of fuel
(YF b
= 0) the adiabatic flame temperature Ta
is calculated from
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CH4 + 15 (0.21O2 + 0.79 N2) �! b1 CO2 + b2 H2O + b3 N2 + b4 O2
atom conservation ) b1 = 1 b2 = 2 b3 = 11.85 b4 = 1.15
Q = ho
CH4� ho
CO2� 2ho
H2O = 191.755 kcal
use an iterative procedure
for Ta = 2000K, LHS = 231.904for Ta = 1700K, LHS = 187.019
this is not accurate because we did
not account for product dissociation.
Z Ta
Tu
CpbdT = Q
Z Ta
Tu
[Cp1+ 2Cp
2+ 11.85Cp
3+ 1.15Cp
4] dT = 191.755 kcal
Ta = 1732K)
)
It’s convenient to rewrite this relation using the the molar heat capacities Cpi.
Z Ta
Tu
cpbdT =
YF u
⌫FWFQ
Z Ta
Tu
CpbdT = Q
Example:
Let Tu = 298K
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CH4 + 15 (0.21O2 + 0.79 N2) �!CO2, H2O, N2, O2, NO, H,
With product dissociation
and the determination of the final compositions requires, in addition to the atom
conservation equations, chemical equilibrium equations.
OH, O, N, CO, O3, NO+, etc..
CO2 H2O 1732.1
CO2 H2O CO 1731.9
CO2 H2O CO O 1731.8
CO2 H2O CO O N NO 1727.3
CO2 H2O CO O N NO, H, OH 1725.7
The inclusion of other products such as O3, CH3, HO2, CH2O, C2H6, NO2,HCO etc., did not change the adiabatic flame temperature.
Ta
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For complete combustion of fuel (YF b= 0)
the adiabatic flame temperature Ta is calculated from
Ta = Tu +(Q/cp)YF u
⌫FWF
Ta = Tu +(Q/cp)YOu
⌫OWO
Z Ta
Tu
cpbdT =
YF u
⌫FWFQ
Assume cp is nearly constant, the adiabatic temperature for a lean mixture,
where the fuel is totally consumed (YF b= 0) is
For a rich mixture, the oxidizer is totally consumed (YOb= 0), and we obtain
in an analogous way
The two expressions are identical when � = 1
Simplified expression
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Combustion28
formation of CO, the major dissociation product of CO 2 , so that the lower the H/C ratio, the higher the temperature. Thus for equivalence ratios around unity and very high energy content, the lower the H/C ratio, the greater the tempera-ture; that is, the H/C curves intersect.
As the pressure is increased in a combustion system, the amount of disso-ciation decreases and the temperature rises, as shown in Fig. 1.6 . This observa-tion follows directly from Le Chatelier’s principle. The effect is greatest, of course, at the stoichiometric air–fuel mixture ratio where the amount of dis-sociation is greatest. In a system that has little dissociation, the pressure effect on temperature is small. As one proceeds to a very lean operation, the tem-peratures and degree of dissociation are very low compared to the stoichio-metric values; thus the temperature rise due to an increase in pressure is also very small. Figure 1.6 reports the calculated stoichiometric fl ame temperatures for propane and hydrogen in air and in pure oxygen as a function of pressure. Tables 1.3–1.6 list the product compositions of these fuels for three stoichi-ometries and pressures of 1 and 10 atm. As will be noted in Tables 1.3 and 1.5 , the dissociation is minimal, amounting to about 3% at 1 atm and 2% at 10 atm. Thus one would not expect a large rise in temperature for this 10-fold increase
TABLE 1.2 Approximate Flame Temperatures of Various Stoichiometric Mixtures, Initial Temperature 298 K
Fuel Oxidizer Pressure (atm) Temperature (K)
Acetylene Air 1 2600 a
Acetylene Oxygen 1 3410 b
Carbon monoxide Air 1 2400
Carbon monoxide Oxygen 1 3220
Heptane Air 1 2290
Heptane Oxygen 1 3100
Hydrogen Air 1 2400
Hydrogen Oxygen 1 3080
Methane Air 1 2210
Methane Air 20 2270
Methane Oxygen 1 3030
Methane Oxygen 20 3460
a This maximum exists at φ ! 1.3. b This maximum exists at φ ! 1.7.
CH01-P088573.indd 28CH01-P088573.indd 28 7/24/2008 2:52:11 PM7/24/2008 2:52:11 PM
Glassman & Yetter, 2008
�1 richlean
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Law, 2006
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If the reference temperature T o
= Tu, for complete combustion of fuel
(YF b
= 0) the adiabatic flame temperature Ta
is calculated from
Assuming constant specific heats
Tb = Tu +
(Q/cv)YF u
⌫FWF, Tb = Tu +
(Q/cv)YOu
⌫OWOfor lean/rich mixtures
Z Tb
T o
cpbdT =
YF u
⌫FWFQ+
RW
(Tb � Tu)
and Tb is clearly larger than the adiabatic flame temperature Ta.
Constant volume equilibrium temperature
If a given combustible mixture is made to approach chemical equilibrium in an
isochoric (constant volume), adiabatic process, then the temperature attained
by the system Tb is higher than the adiabatic flame temperature.
eu = eb
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Simplified Model and
Dimensionless Equations
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⌫F Fuel + ⌫O Oxidizer ! Products
In the following the chemistry will be represented by a global one-step (irre-
versible) reaction
describing the combustion of a single fuel.
! = B✓⇢YF
WF
◆nF✓⇢YO
WO
◆nO
e�E/RT
The reaction will be assumed of order nF , nO, with respect to the fuel/oxidizer,
and an overall order n = nF + nO. The reaction rate will be assumed to obey
an Arrhenius law
with an overall activation energy E and a pre-exponential factor B (treated as
constant).
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p = �RT/W
�Dv
Dt= �⇥p+ µ
⇥⇥2v + 1
3⇥(⇥ · v)⇤+ �g
⇥�
⇥t+� · �v = 0
⇥cpDT
Dt� �r2T =
Dp
Dt+ �+Q⇤
! = B✓⇢YF
WF
◆nF✓⇢YO
WO
◆nO
e�E/RT
⇢DYi
Dt� ⇢Dir2Yi = �⌫iWi !, i = F,O
For simplicity, we will treat µ,�, ⇢Di constants (although some of the theoreti-
cal development could accommodate temperature-dependent transport without
much di�culty) so that
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Non-dimensional Equations
Characteristic values:
• the fresh unburned state p0, ⇢0, T0 (satisfying p0 = ⇢0RT0/W ) for p, ⇢, T
• a characteristic speed v0 to be specified
• the di↵usion length lD ⌘ Dth/v0 for distances, where Dth = �/⇢cp is the
mixture thermal di↵usivity
• the di↵usion time lD/v0 for t
This choice is clearly not unique and there may be other length, time, and
velocity scales that, for a given problem, could be more relevant.
We will use the same variables for the dimensionless quantities; i.e. after sub-
stituting
˜v = v/v0 say, we remove the .
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⇥�
⇥t+� · �v = 0
⇥Dv
Dt= � 1
�M2⇥p+ Pr
⇥⇥2v + 1
3⇥(⇥ · v)⇤+ Fr�1⇥eg
�DYF
Dt� Le�1
F r2YF = �⇥
⇥DYO
Dt� Le�1
O r2YO = ��⇤
p = �T
! = D ⇢nY nF
F Y nO
O e��0/T
⇢DT
Dt�r2T =
��1
�
✓Dp
Dt+ � PrM2 �
◆+ q !
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Dimensionless Parameters
M =v0p
�p0/⇢0Mach number Fr =
v20/lDg
Froude number
Lewis numberLei =�/⇢cpDi of species i
Pr =µcp�
Prandtl number
Note that Pr
�1=⇢0v0lD/µ = Re where Re is
the Reynolds number based on the di↵usion length.
The Reynolds number based on a hydrodynamic
length L will be large because typically L � lD.
mass weighted
stoichiometric coe↵.
⌫ =⌫OWO
⌫FWF
Damkohler number
=
flow time
reaction time
D =(lD/v0)
[(⇢0/WF )nF�1(⇢0/WO)
nO ⌫FB]�1
�0 = E/RT0activation energy
parameter
q =Q/⌫FWF
cpT0
heat releaseparameter
numerator is the heat release
per unit mass of fuel
Note: the units of ⌫FB, like the units of k is [(conc)
n�1(time)]
�1(one usually
set ⌫F = 1 in writing the chemical reaction equation).Moshe Matalon
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Low Mach Number Approximation
The propagation speed of ordinary deflagration waves is in the range 1-100
cm/s, namely much smaller than the speed of sound (in air a0 = 34, 000 cm/s).
M ⌧ 1
momentum equation ) rp = 0
p = P (t) + �M2 p(x, t) + · · ·
will all other variables expressed as v + �M2ˆv + · · ·
⇢DT
Dt�r2T =
��1
�
dP
dt+ q !
⇢Dv
Dt= �rp+ Pr
⇥r2v + 1
3r(r · v)⇤+ Fr�1⇢eg
⇢T = P (t)
acoustic disturbances travel infinitely fast, and are filtered out.
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Unless P (t) is specified, we are missing an equation, since p has been replaced
by two variables P and p. An equation in bounded problems can be obtained
as follows:
⇢@T
@t+ ⇢v ·rT �r2T =
��1
�
dP
dt+ q !
@(⇢T )
@t+r · (⇢vT )�r2T =
��1
�
dP
dt+ q !
1
�
dP
dt= �r · (Pv �rT ) + q!
1
�
Z
V
dP
dtdV = �
Z
S
(Pv�rT ) · n dS + q
Z
V
!dV
on the surface S, v · n = 0 and for
adiabatic conditions @T/@n = 0.
dP
dt=
�q
V
Z
V
! dV
@⇢
@t+r · ⇢v = 0 / T
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The low Mach number equations are (with the ”hat” in p removed), therefore
⇥�
⇥t+� · �v = 0
�DYF
Dt� Le�1
F r2YF = �⇥
⇥DYO
Dt� Le�1
O r2YO = ��⇤
⇢Dv
Dt= �rp+ Pr
⇥r2v + 1
3r(r · v)⇤+ Fr�1⇢eg
⇢DT
Dt�r2T =
� � 1
�
dP
dt+ q !
and when the underlying pressure does not change in time, P = 1.
⇢T = P
Otherwise, unless it is specified, we need an equation for P (t) (the pressure phas been replaced by two variables P and p), which can be obtained by a globalintegration across the entire volume (as discussed below).
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⇢DT
Dt�r2T = q !
�DYF
Dt� Le�1
F r2YF = �⇥
⇥DYO
Dt� Le�1
O r2YO = ��⇤
⇢DHi
Dt�r2Hi = 0
leaving only one equation with the highly nonlinear reaction rate term. This is
a great simplification, but as we shall see, small variations of the Lewis numbers
from one produce instabilities and nontrivial consequences.
The combinations HF = T +qYF and HO = T +q YO/⌫ (and hence YF �YO/⌫)satisfy reaction-free equations
For unity Lewis numbers the operator
on the left hand side of these three equations
is the same.
Coupling Functions
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⇥�
⇥t+� · �v = 0
⇢(@v
@t+ (v ·r)v) = �rp+ Pr
⇥r2v + 1
3r(r · v)⇤+ Fr�1⇢eg
solve to determine v
with the given v solve for T, YF , YO.
The constant-density approximation
Thermo-Di↵usive model
⇢ = 1
⇢(@YF
@t+ v ·rYF )� Le�1
F r2YF = �!
⇢(@YO
@t+ v ·rYO)� Le�1
O r2YO = �⌫!
⇢(@T
@t+ v · T )�r2T = q !
Can be derived systematicallyby assuming that the heat release q ⌧ 1
⇢T = 1One must abandon the equation of state
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