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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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Copyright  ©2011  by  Moshe  Matalon.  This  material  is  not  to  be  sold,  reproduced  or  distributed  without  the  prior  wri@en  permission  of  the  owner,  M.  Matalon.   3  

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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