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TRANSCRIPT
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Lecture 3 Stoichiometry & Adiabatic Flame Temperature
Transport properties
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N�
i=1
ν�iMi →N�
i=1
�i Mi
We have seen that the net production/consumption (moles per unit volume persecond) for a global one step reaction
is given by
ωi = (ν��i −ν�i)Wi ω i = 1., 2 . . . , N
the term involving the net rate of production of species i appearing in the energy
and species equation, ωi , denotes the time rate of change of the mass of species
i (grams, per unit volume per second). Hence ωi = Wiω̂i, and
N�
i=1
ωi = ωN�
i=1
(ν��i − ν�i)Wi = 0note that
ω̂i = (ν��i −ν�i)ω = (ν��i −ν�i) kN�
i=1
Cν�i
i i = 1., 2 . . . , N
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N�
i=1
ωihoi = ω
N�
i=1
(ν��i −ν�i)Wi hoi
� �� �−Q
= −Qω
ωi = (ν��i −ν�i)Wi ω i = 1., 2 . . . , N
Hence
and the term appearing in the energy equation becomes
ω = BTαN�
i=1
�ρYi
Wi
�ν�i
e−E/RT
difference between the heat of formationof the products and reactants
(negative for exothermic reactions)is the total heat released, orthe heat of combustion
with ν�i in the exponents may be replaced with empirical ni, as necessary.
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N�
i=1
ν�iMi →N�
i=1
�i Mi
dni
ν��i − ν�i=
dnj
ν��j − ν�j
or in terms of the partial massesdmi
(ν��i − ν�i)Wi=
dmj
(ν��j − ν�j)Wj
dYi
(ν��i − ν�i)Wi=
dYj
(ν��j − ν�j)Wj
since the total of mass (unlike the total number of moles) in the system isunchanged by chemical reaction
Stoichiometry
Given the reaction
we have seen that there is a relation between the change in the number of molesof each species; i.e., for any two species i and j
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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
νFWF
=dYO
νOWO
νF Fuel + νO Oxidizer → Products
ν�F CmHn + ν�O O2 → ν��CO2CO2 + ν��H2O
H2O
with the stoichiometric coefficientsν�F = 1, ν�O = m+ n/4, ν��CO2
= m, �H2O= n/2
where ν�F was taken equal to one, arbitrarily.
which may be written as (primes are unnecessary)
then νO/νF is the ratio of the stoichiometric coefficients, and
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dYF
νFWF
=dYO
νOWO
YF − YF u
νFWF
=YO − YOu
νOWO
A fuel-air mixture is called stoichiometry, if the fuel-to-oxygen ratio is suchthat both reactants are entirely consumed when combustion to CO2 and H2Ois completed.
YOu
YF u
�����st
=νOWO
νFWF
≡ νmass-weightedstoichiometric ratio
between the initial unburned state (subscript u) and a later state
Integrating
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φ =YF u/YOu
YF u/YOu
��st
=YF /YO
νFWF /νOWO
=νYF u
YOu
In general, the initial state may not be at stoichiometry. A measure of thedeparture from stoichiometry is given by the equivalent ratio φ
0 < φ < ∞
for combustion in air, the equivalence ratio is often expressed as the fuel-to-airratio
φ = 1 stoichiometric mixtureφ < 1 lean mixture (in fuel)φ > 1 rich mixture (in fuel)
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Adiabatic Flame Temperature
If a given combustible mixture is made to approach chemical equilibrium bymeans of an isobaric, adiabatic process, then the temperature attained by thesystem is the adiabatic flame temperature Ta.
For an adiabatic, isobaric process dh = 0. Integrating from the unburned to theburned state
hu = hb
N�
i=1
Yiuhi =N�
i=1
Yibhi
N�
i=1
Yiuhoi +
� Tb
T o
cpbdT =
N�
i=1
Yibhoi +
� Tu
T o
cpudT
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N�
i=1
�Yiu − Yib
�hoi =
� Tb
T o
cpbdT −
� Tu
T o
cpudT
cpu=
N�
i=1
Yiucpi(T ) , cpb=
N�
i=1
Yibcpi(T )
where the specific heats are those of the mixture, calculated with the massfractions of the unburned/burned gas, respectively
dYi
(ν��i − ν�i)Wi=
dYj
(ν��j − ν�j)Wj
can be integrated to give
For a one-step global reaction, where we consider j = 1 to be the fuel, denotedby F, the equation
Yiu − Yib =�YF b
− YF u
� (ν��i − ν�i)Wi
νFWF
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N�
i=1
�Yiu − Yib
�hoi
� �� �=
� Tb
T o
cpbdT −
� Tu
T o
cpudT
YF b− YF u
νFWF
N�
i=1
(ν��i − ν�i)Wihoi
� �� �−Q
�
� Tb
T o
cpbdT −
� Tu
T o
cpudT =
YF u− YF b
νFWFQ
� Ta
T o
cpbdT =
YF u
νFWFQ
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 = hoCH4
− hoCO2
− 2hoH2O = 191.755 kcal
Example: calculate the adiabatic flame temperature for the combustion ofmethane/air given by CH4 + 15 (0.21O2 + 0.79 N2) at 298K
Using the molar heat capacities Cpi,the adiabatic flame temperature Ta is calculated from
� Ta
T o
CpbdT = Q
use an iterative procedure
� Ta
T o
[Cp1+ 2Cp
2+ 11.85Cp
3+ 1.15Cp
4] dT = 191.755 kcal
for Ta = 2000K, LHS = 231.904for Ta = 1700K, LHS = 187.019 ⇒ Ta = 1702K
this is not accurate because we didnot account for product dissociation.
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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 atomconservation equations, chemical equilibrium equations.
OH, O, N, CO, O3, NO+, etc..
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For a rich mixture, the oxidizer is totally consumed (YOb= 0), and we obtain
in an analogous way
Ta = Tu +(Q/cp)YOu
νOWO
Assume cp is nearly constant, the adiabatic temperature for a lean mixture,where the fuel is totally consumed (YF b
= 0) is
Ta = Tu +(Q/cp)YF u
νFWF
� Ta
T o
cpbdT =
YF u
νFWFQ
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Law, 2006 6/26/2011
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The bulk viscosity is negligible in combustion processes and will be set to zero,i.e., κ = 0.
The dependence of the binary diffusion coefficients Di (which actually standsfor Di,N ) on pressure and temperature is
Di ∼ Tα/p 3/2 ≤ α ≤ 2
Typical values at p = 1atm are in the range 0.01− 10 cm2/s.
Although the transport coefficients λ, µ in a multicomponent mixture depend ingeneral on the concentrations, this dependence is generally ignored and averagequantities are used; they remain dependent on pressure and temperature.
Similarly W is assumed constant.
Transport Coefficients
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The thermal conductivity depends mostly on temperature, and behaves as
λ ∼ Tα/p 1/2 ≤ α ≤ 1
More relevant, however, is the thermal diffusivity λ/ρcp which has the same unitsand the same temperature and pressure dependence as the diffusion coefficientsand the kinematic viscosity. In particular
λ/ρcp ∼ Tα/p 3/2 ≤ α ≤ 2
Typical values at p = 1atm are in the range 0.1− 1 cm2/s.
The kinematic viscosity µ/ρ has the same units and the same pressure andtemperature dependence as the binary diffusion coefficients
µ/ρ ∼ Tα/p 3/2 ≤ α ≤ 2
Typical values at p = 1atm are in the range 0.1− 1 cm2/s.
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Since λ/ρcp, µ/ρ,Di have the same dependence on temperature, their ratios arenearly constant.
λ/ρcpDi
= Lei
µ/ρ
λ/ρcp= Pr
µ/ρ
Di= Sci
Lewis number
Prandtl number
Schmidt number
We can write ρDi = Le−1i (λ/cp) and µ = Pr(λ/cp) with λ = λ(T ). A good
approximation (Smooke & Giovangigli, 1992) for the latter is
λ
cp= 2.58 · 10−4 g
cm s
�T
298K
�
The Prandtl and Lewis numbers are nearly constant; Pr = 0.75 and Le variesin the range 0.2− 1.8.6/26/2011
νF Fuel + νO Oxidizer → Products
We will be mostly concerned with a global reaction
ω = BTα
N�
i=1reactants
�ρYi
Wi
�ni
e−E/RT = BTα
�ρYF
WF
�nF�ρYO
WO
�nO
e−E/RT
with α = 0 (although in the asymptotic treatment one can easily retain thisfactor with minor modifications).
ω = B�ρYF
WF
�nF�ρYO
WO
�nO
e−E/RT
n = nF + nO is the (overall) reaction ordernF , nO are the reaction orders w.r.t. the fuel/oxidizer
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