notes lecture 5 - cefrc.princeton.edu

Summer 2013

Moshe Matalon University of Illinois at Urbana-‐Champaign 1

Lecture 5

Stoichiometry Dimensionless Parameters

Moshe Matalon

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

Moshe Matalon

Summer 2013


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

Moshe Matalon

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.

Moshe Matalon

Summer 2013


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

Moshe Matalon

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

Moshe Matalon

Summer 2013


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

Moshe Matalon

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

Moshe Matalon

Summer 2013


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

Moshe Matalon

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

Moshe Matalon

Summer 2013


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

Moshe Matalon

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

Moshe Matalon

Summer 2013


Law, 2006

Moshe Matalon

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

Moshe Matalon

Summer 2013


Simplified Model and

Dimensionless Equations

Moshe Matalon

⌫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).

Moshe Matalon

Summer 2013


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

Moshe Matalon

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 .

Moshe Matalon

Summer 2013


⇥�

⇥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 !

Moshe Matalon

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

Summer 2013


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.

Moshe Matalon

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

Moshe Matalon

Summer 2013


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).

Moshe Matalon

⇢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

Moshe Matalon

Summer 2013


⇥�

⇥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

Moshe Matalon

notes lecture 5 - cefrc.princeton.edu

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