Lecture 9

• CFD modeling of combustion – 9.a

• Basic combustion concepts

– 9.b

• Governing equations for reacting flow

• Reference books

– An introduction to computational fluid dynamics, the finite volume method, H.K. versteeg, W.

Malalasekera

• Chapter 12

– Theoretical and numerical combustion, (2nd edition) , T. Poinsot, D. Veynante,

• Chapter 1

1

A few examples of combustion

2

Keywords:

Fire, power, Heat, light, color, emission, pollution,

Chemical reactions, multi-component mixture, radicals,

Flame, combustion acoustic, unstable combustion, detonation, etc.

fuel oxidizer products

� > 0

heat

Combustions

3

Combustion usually takes place in gas-phase,

through certain exothermic chemical reaction a cold

fuel/oxidizer mixture is turned into a hot product

mixture, a sustained combustion process happen in

a non-stationary flow environment which heats up a

continuous supply of freshly mixed fuel and oxidizer

gases.

Physical conservation laws:

Conservation of mass (for each atom element)

Conservation of momentum (|�| > 0).

Conservation of energy.(Heat, mechanical work, kinetic energy, etc.)

A first observation:

A reacting flow domain can be regarded

as a multi-component gas “mixture”

composed of different species.

Relevant concepts to describe a muti-component gas mixture

4

��� �

�

How much percentage a certain species of � is inside a mixture?

Mole (number) fraction: � → ��∑ ����

, � >> 1Mass fraction: � → ����

∑ �������: mean molecular weight of the mixture

� : mole weight for species k

�: density of mixture

� : for species k

�: pressure of mixture

� : for species ���: specific Heat capacity at constant volume for mixture

��, for species ���: specific heat capacity at constant pressure for mixture

��, for species �ℎ: Total Enthalpy of mixture

ℎ for species �ℎ� : Sensible Enthalpy of mixture

ℎ�, for species �Δℎ!": Enthalpy of formation

Δℎ!, " for species �

��� + → � + ��: 1 2 3 4

A combustion mixture contains multiple species () ≥ 1)The mass fractions � , the mole fractions � for each species � ∈ (1, … . , ))

5

��� + 2 → � + 2�

Mass fraction for species � : � ∈[0,,1,�] = [Y56�, �7 , �87, �97]

Mole fraction for species � : � ∈[0,,1,�] = [X56�, �7 , �87, �97]

Molecular weight for species �:

� ∈[0,,1,�] = [�89�, �7, �87, �97] = [12 + 6, 12 × 2, 12 + 16 × 2, 1 × 2 + 16] gram/mole

� = � �

/ > � ?� ?

@

A= �

���

> � = 1 @

� = � � / > � A� A@

A= �

���

� = 1/ > � �

= > � � @

@

Mixture-averaged mean

molecular weight

> � = 1 @

Total pressure, partial pressure, Equation of statefor a mixture containing multiple species � ∈ (1, … . , ))

6

�0B = �0CD … � B = � CD

… �@B = �@CD

> � B =@

E0> � CD

@

E0

� = ∑ � @ E0B CD = �CD/�

� = > � @

E0Partial pressure and

equation of state for a

single species

� = � �

CD

Total pressure:

Equation of state

for the mixture

Mean molecular weight

C: universal gas const.

� = ��

� ≡ ��

� = 1/ > � �

@

Thermodynamics: Enthalpy and internal energy in a single-species system

7

�

First law of thermodynamics (conservation of energy)

Δ Z⏞internal energy

= \⏞]^_`

− �⏞bcd

Internal energyenergyenergyenergy: e(i) “Sensible” energy (averaged “kinetic energy” of the random moving moleculars)(ii) “chemical, or formation” energy stored in chemicalchemicalchemicalchemical----bondsbondsbondsbonds

Enthapy h= Z + �B: At constant pressure system (Volume change)

Δ ℎ⏞q�`]_r�s

= Δ Z + Δ �B → Δ Z + pΔB tbcd

constant pressure

(uvw�Z × xyz{|��Z) vw (uvw�Z} × Bv~�}Z)

Total enthalpy, sensible enthalpy and enthalpy of formation for a specie �

8

ℎ = ℎ�, + Δℎ!, "

= � ��, xD�

��+ Δℎ!, "

ℎ : Enthalpy [�

�] of a species (k) with respect to reference enthalpy at standard

conditions at pressure (1ATM) and temperature (D"=298.15K)

Sensible enthalpy: ℎ�, chemical , enthalpy of formation (

� �),

Enthalpy of formation Δℎ!, " : increase in enthalpy when a compound is

formed from its constitute elements in their nature forms at standard

conditions, for H2, O2 , N2, C (graphite) it is zero, for � it is -393 520

KJ/kmol, because the exothermic reaction(heat release):

�(�w|�ℎy{Z) + ��

Total = sensible +chemical

��, : specific heat capacity at constant pressure for species k

Sensible energy and chemical energy for a single specie �

9

Z = ℎ − � �

= ℎ�, − � �

+ Δℎ!, "

= � ��, xD�

��− CD"

� + Δℎ!, "

= Z�, + Δℎ!, "

Sensible

energy Z�, Chemical, enthalpy

of formation (�

�)

Sensible+chemical energy

��, : specific heat capacity at constant volume for species �

Enthalpy and Energy in a multi-component mixture

10

ℎ = ∑ � ℎ @ E0 = ∑ � � ��, xD + Δℎ!, "���

� = � ∑ � ��, � xD + ∑ � Δℎ!, "�

���

= � �� xD ��� + ∑ � Δℎ!, "�

Z = ∑ � Z @ E0 = ∑ � � ��, xD��� − ���

��+ Δℎ!, "�

= � ∑ � ��, � xD��� − CD" ∑ ��

��� + ∑ � Δℎ!, "�

= � �� xD��� − CD"/� + ∑ � Δℎ!, "�

Enthalpy of Mixture:

Energy of Mixture:

�� & ��: Mixture-averaged heat capacity at constant volume and pressure respectively

��, & ��, : Heat capacities for a single spices �

Enthalpy of formation for mixture

Relation between Energy and enthalpy for a mulit-component mixture and for each single species

11

Z = ℎ − � �

Z = > � Z @

= > � ℎ − �

�

@

= ℎ − �

�

Apply first law of thermodynamics to an adiabatic (0D) combustion problem

12

Assume homogenous (no spatial gradient), zero mean velocity, adibatic

To find Burned (product) state D�, �� with mass fraction � � = Y56�� , �7� , �87� , �97� ?

� ���xD + > � �Δℎ!, "@

��

��

]�

= � ���xD + > � �Δℎ!, "@

��

��

]�

Given Unburned (fresh) state D�, ��,with mass fraction � � = [Y56�� , �7� , �87� , �97� ]

�� = ��constant pressure

constant volume �� = �� � ���xD

��

��− CD"

�� + > � �Δℎ!, "@

^�

= � ���xD��

��− CD"/�� + > � �Δℎ!, "

@

^�

?� �

Z� = Z�

ℎ� = ℎ�

�� = ��/� CD�

Example: Assume a global, single-step, irreversible reaction,

determine the final burned mass fraction � �

13

1 ⋅ ��� + 2 ⋅ ⇒ 1 ⋅ � + 2 ⋅ � 1Δ ∶ 2Δ ⇒ 1Δ ∶ 2Δ

� � = [Y56�� , �7� , �87� , �97� ] � � = [X56�� , �7� , �87� , �97� ]

� �∗ = � � + �

� � = � �∗

∑ � ?�∗@ ?E0= � � + �

1 + ∑ � ?@ ?E0

� � � �

∑ � �∗@ E0 = 1 + ∑ � @ E0 ≠ 1 ,

normalize � �∗

to get mole

fraction of burned state � �mole fraction of burned state!

mole fraction of unburned statemass fraction of unburned state

Assumption: either fuel or oxidizer

must be completely consumed.

Δ = min ( �89� � , 0 �7 � )

� = (� AA − � A ) ⋅ Δ(�)

Left

coeff

�′ Right Right Right Right coeff�′′

(1)��� 1 0(2) 2 0(3)� 0 1(4)� 0 2

Given

Reactions conserve atomic elements

Some basic concepts relevant for combustion chemical reactions

14

��� �

� Chemical reactions The reaction mechanism

Globally reduced reaction

Stoichiometry/ Equivalence ratio

Detailed reaction mechanism

Elementary reactions

Unimolecular, Bimolecular and Termolecular

Reaction pathway

Intermediate species

Reversible reactions and chemical equilibrium

Finite rate of chemical reaction

Reaction rate constant

Arrhenius law

Activation energy.

The globally reduced single-step chemical reaction systemDifferent ways of preparing the reactant-mixture

15

1 ⋅ ��� + 2 ⋅ ( ) ⇒ 1 ⋅ � + 2 ⋅ �

Conservation of each element:

> �′ �

�}

[5,�� 6,�,�] = > �′′ �

�}

[5, �� 6,�,�]

} [8]

: number of a element [C] contained

within the molecular of species �

1 ⋅ ��� + 3 (+3.76)) ¡d

⇒ 1 ⋅ � + 2 ⋅ � + 2 ⋅ 3.76) + 1 (+3.76)) ¡d

1 ⋅ ��� + 3 (+3.76)) ¡d

⇒ 1 ⋅ � + 2 ⋅ � + 3 ⋅ 3.76) + 1 ⋅

1 ⋅ ��� + 2 ⋅ (+3.76)) ¡d

⇒ 1 ⋅ � + 2 ⋅ � + 2 ⋅ 3.76)

1 ⋅ ��� + 3 (+3.76)) ¡d

+ � ⇒ 2 ⋅ � + 2 ⋅ � + 3 ⋅ 3.76) + 1 ⋅

stoichiometry

stoichiometry

1 ⋅ ��� + 12 (+3.76))

 ¡d ⇒ 1

4 � + 12 � + 1

2 ⋅ 3.76) + 34 ���

(�)Left

�′ RightRightRightRight�′′

(1)��� 1 ¾(2) ½ 0

(3)� 0 ¼(4)� 0 ½(5)) 3.76/2 3.76/2

(�)Left

�′ RightRightRightRight�′′

(1)��� 1 ¾(2) 0 0

(3)� 0 ¼(4)� 0 ½(5)) 0 3.76/2

(6) Air ½ 0

Lets examine a global, single-step, fuel+oxidizer reaction system

Stoichiometry and equivalence ratio

16

1 ⋅ ���t¥�^r

+ 2 (+3.76)) ¡d

⇒ 1 ⋅ � + 2 ⋅ � + 2 ⋅ 3.76)1Δ ∶ 2Δ ∶ 1Δ ∶ 2Δ ∶ 2 ⋅ 3.76Δ

Both fuel and oxidizer are

completely consumed!

Δ|¦§=�89� � = 0 � ¡d�

�¥�^r�7¨¡©¡ª^d �`

= �′¥�^r�′7¨¡©¡ª^d

= 12

�¥�^r�7¨¡©¡ª^d �`

= �′¥�^r ⋅ �¥�^r�′7¨¡©¡ª^d ⋅ �7¨¡©¡ª^d

= 1 ⋅ �89�2(W�¬ + 3.76W@¬)

Equivalence ratio: ­ =�®�¯°

�±²³´³µ¯¶ _·`�_r / �®�¯°

�±²³´³µ¯¶ �`­ > 1: ¸�Z~ wy�ℎ­ < 1: ¸�Z~ ~Z|�

­ = 1: º{vy�ℎyv}Z{w»1 ⋅ ��� + 3 ⋅ ( +3.76)) ­ = 1 ⋅ �¥�^r

3 ⋅ � ¡d/ 1 ⋅ �¥�^r

2 ⋅ � ¡d= 2

3 < 1:¸�Z~ ~Z|�

1 ⋅ ��� + 3 ⋅ ( +3.76)) ⇒ 1 ⋅ � + 2 ⋅ � + 2 ⋅ 3.76) + 1 ⋅ ( + 3.76))No fuel or oxidizer coexist

on the product side

z{

Estimation of adiabatic flame temperature

17

If a fuel/oxidizer mixture is burned completely (assume under constant pressure), and if no external heat

or work transfer takes place , then all energy liberated by chemical reaction will heat the product,

achieving max (adiabatic ) flame temperature!

� � → � �

� �∗

normalize→ � �

Δ = mi� ( �89� � , 0 � ¡d� )

� ���xD + > � �Δℎ!, "@

��

��= � ���xD + > � �Δℎ!, "

@

��

��

� � → � �

D�

� � ∈ [Y56�� , �7� , �@� , �87� = 0, �97� = 0]

Note: for non-stoichiometry mixture (i.e. ­ ≠ 1), the product mixture

may contain unburned fuel or oxidizer (i.e. �!�^r� ≠ 0 or �c¨¡©¡ª^d� ≠ 0)

stoichiometry 1 ⋅ ��� + 2 ⋅ ( +3.76)) ⇒ 1 ⋅ � + 2 ⋅ � + 2 ⋅ 3.76) 1Δ ∶ 2Δ ∶ 2 ⋅ 3.76Δ ∶ 1Δ ∶ 2Δ ∶ 2 ⋅ 3.76Δ

� = (� AA − � A ) ⋅ Δ(�)

Left �′ RightRightRightRight�′′

(1)��� 1 0(2) 2 0

(3)� 0 1(4)� 0 2(5)) 2 ⋅ 3.76 2 ⋅ 3.76

� �∗ = � � + �

Reaction

� � → � �

Chemical equilibrium and reverse reaction

18

In practice, some reactions occur in the reverse direction (more prominent at high temperature).

� ⇌ � + 12

� ⇌ � + 12

� ⇌ � + �� ⇌ � + �

…

� = ℎ − D ⋅ z specific entropy z: � �⋅ ½Gibbs function [ �

�]: Equilibrium maximize Gibbs function

�  ⋅ ¾ + �� ⋅ ¿ + �· ⋅ � + ⋯ ⇌ �^ ⋅ Á + �! ⋅ u + ⋯

Condition for equilibrium: ΔÂ�" = −CD log Ã�

Ã� = �q�¥ …� �Ä�8 . . . = �^�! …

�_���· . .Equilibriums constant.

Combustion: chemical reaction mechanismExample of hydrogen oxidization

19

A globally reduced one-step reaction � + 12 ⇒ �

A detailed reaction mechanism contain multiple elementary reactions involving

many intermediate species

� + ⇌ 2� � + ⇌ � + �

� + � ⇌ � + �� + � ⇌ � + �

� + ⇌ � + � + ⇌ � + �

� + � + Å ⇌ � + Å….

�, �, , �,intermediate species (radicals), Å denotes third body ( or,

arbitrary atom/radical/molecures which increase the collision chance for

chemical reactions)

Detailed chemistry, Intermediate species

20

Another example for methane oxidization

Detailed chemistry, Intermediate species

Example for methane oxidization

21

A detailed GRI-mechanism(still not complete) contains 325 elementary

reactions, 53 species, which is optimized for certain ranges of

temperature and pressure conditions.

Different chemical

reaction “pathway”

or subsystem.

Chemical reaction does not happen in an instant, it takes time…Elementary reactions and the reaction rate

22

Molecularity Elementary Step Rate Law for Elementary step [ Æcr

ÆÇ⋅�]

Unimolecular ¾ ½→ �wvx��{z w|{Z = Ã[¾]Bimolecular ¾ + ¿ ½→ �wvx��{z w|{Z = Ã[¾][¿]

¾ + ¾ ½→ �wvx��{z w|{Z = à ¾

Termolecular ¾ + ¾ + ¿ ½→ �wvx��{z w|{Z = à ¾ [¿]¾ + ¾ + ¾ ½→ �wvx��{z w|{Z = à ¾ 1

¾ + ¿ + � ½→ �wvx��{z w|{Z = Ã[¾][¿][�]

¾ + ¾ + ¿Ã0⇌

ÃÈ0� + É

w|{Z = Ã0 ¾ ¿wate = ÃÈ0[�] É

¾ + ¾ + ¿ ½Ê � + É

� + É ½ËÊ ¾ + ¾ + ¿

[¾] : Æcr

ÆÇ

Note: forward/backward reaction can also be related through equilibrium condition

Reaction rate constant and Arrhenius law

23

Ã(�) = ¾DÌexp (− Á_CD)

Reaction rate constant :

(Arrhenius law)

¾: pre-exponential constant

Î : temperature exponent

Á_: Activation energy.

Just a note: Ã has different unit for

different order of elementary reaction

Unimolecular , w|{Z = Ã[¾]Bimolecular , w|{Z = Ã[¾][¿]

..

à → 0 when D ≪ D_ ≡ qÐ�à ≫ 0 when D ≫ D_

Determine the reaction rate of a specie Ò involved in multiple Ó elementary reactions

24

1: … ½Ê …Ô: 1¾ + 0¿ + ⋯ + 2� + ⋯ ½¬ 0¾ + 1¿ + ⋯ + 0� + ⋯ … …→ …

Ó: 0¾ + 2¿ + ⋯ + 1� + ⋯ ½Õ 2¾ + 0¿ + ⋯ + 0� + ⋯… …→ …

All elementary reactions (all rewritten as forward reaction)

Ö×Ø = > Ö×Ø,ÙÚÙE0

All species

1: ¾2: ¿

…Ò: �

…):…

) Û

ÃÙ = ¾ÙDÌÕexp (− Á_ÙCD )

Total mole concentration

Ö×Ø,Ù = (�Ø,ÙAA −�Ø,ÙA ) ÃÙ ∏ �¡� ݳ,Õ?@¡E0

Þ× Ø = �Ø ÖØ× , ∀ k = 1, … )rate in mass

unit

rate in mole

unit

Governing equations describing temporal evolution for a

(homogenous, adiabatic, stationary) reacting mixture

xx{ ℎ = 0

) + 2 Unknowns for the above ) + 2 equations:

à á = [ �({), D({), �Æ { , � = 1, … ) ]Initial conditions:

à {" = [ �({"), D({"), �Æ {" , � = 1, … ) ]

∑� = 1©©` � = Þ× , � = 1, … ) − 1

� = �CD/�

xx{ Z = 0

Const. pressure Constant volume+ +or

The process of combustion chemical reaction can be viewed as a (nonlinear) dynamic system problemTypical features in terms of trajectory and attractors for gas phase combustion system

26

A set of ÉÁ equations solved for D({),� { , � ({); k=1,..,N), starting at { = 0.

©©` �0 = Þ0(�0, �, ..,,�@, �, D)©©` � = Þ(�0, �, …, �@, �, D)… ©©` D = Þ�(�0, �,…, �@, �, D)

The solution to the ODEs is a trajectory in high dimensional

phase space spanned by N+2 unknowns variables. A few simple algebraic constraints such as conservation of elements

and also total mass can reduce the number of unknowns.

Combustion chemical reaction can be viewed a (nonlinear) dynamic system problemTypical features in terms of trajectory and attractors for gas phase combustion system

27

©©` Ö0 = Þ0(Ö0, Ö, Ö1)©©` Ö = Þ(Ö0, Ö, Ö1)©©` Ö1 = Þ1(Ö0, Ö, Ö1)

Assume a reduced combustion

system of only three unknowns, the

solution for this nonlinear ordinary

differential equations (ODES) are

trajectory moving in a 3D phase

space spanned by (�0, �, �1).

Initial slow incubation to prepare radical pools

and heat required for “activating” reaction

rapid state change due to large |Þ| cause by à Dafter reaction liberated heat raising temperature

Slowly approaching certain attracting

manifold formed by, for instance, hemical

equilibrium states

Ö0

Ö

Ö1

Catalyst “drill” a tunnel

Þ ∼ Ã � � …

A sketch showing numerical time advancement from three

different initial state points

A note from chemistry

28

Certain (non-gas-phase-combustion) chemical reaction do not have to be

attracted to the equilibrium solution! Their attracting manifold may be a limit-

cycle or even chaotic orbit.

YouTube showing Belousov-zhabotinsky reaction!

https://www.youtube.com/watch?v=0Bt6RPP2ANI#t=00m34s

The Belousov-zhabotinsky reaction!

�0

�

�1

Note: there exist more complicate “attracting manifold” for other nonlinear dynamical system

29

The famous 3D

“butterfly” trajectory with

“chaotic attractor” for

the Lorenze equations

Complex phenomena exists in other

nonlinear dynamics system (Examples:

pendulum system, three-body problem, …)

Combustion equations

©©` Ö0 = Þ0(Ö0, Ö, Ö1)©©` Ö = Þ(Ö0, Ö, Ö1)©©` Ö1 = Þ1(Ö0, Ö, Ö1)

Theoretic and numerical aspects for combustion chemical reaction

30

1) For most gas-phase combustion, there often exists fast and slow reactions, the time scales

of these reactions may differ in several order of magnitude. It is a mathematical “stiff”

system with significantly different time-scales, an expensive adaptive-time-step ODE solver

must be used to perform numerical time-integration.1) Such calculation will usually be performed by “popular” software package: such as Chemkin(free

before, not any more), Cantera (free) and Flamemaster … . Note, accurate calculation of

thermodynamic and transport coefficients (��, , Δℎ", ,..ÉÙ, ) are usually based on the NASA

polynomials, the chemical kinetic mechanism including all elementary reactions and the reacting

constants can be downloaded together with a published journal article.

2) For common gas combustion reaction, there often exist certain “intrinsic lower-dimentional

manifolds” (ILDM) in the phase space, towards which a trajectory will be quickly attracted.

When the trajectory come close to the vicinity of such “manifold” region, the solution along

trajectory then stay parallel and move slowly within such “manifold”.

3) Very expensive calculations of stiff-ODE solver for every CFD-cells.

Ideal: Tabulation

The In-situ adaptive-tabulation (ISAT), by S.B. Pope.

CFD modeling of combustion Governing equations for reacting flow

31

Governing equation for reacting flow

32

å�å{ + å��¡

åæ¡= 0

å��¡å{ + å��¡�Ù

åæÙ= − å�

åæ¡+ åç¡Ù

åæÙ+ � > � ̧,Ù

@

E0

Global Mass

Momentum

ç¡Ù = è(é�³é¨Õ

+é�Õ騳

) − 1 �¡Ù

��

å�¡åæ¡

= − 1�

å�å{ + �¡

å�åæ¡

≠ 0

Burning liberated heat

causes flow dilatation

Combustion does not create

new mass, it just redistributes

mass among different species.

Typical combustion causes ê�ê�

= ���¶ë��

= 5 vw 10 large variation in

dynamic viscosity è(D) and large dilatation term

Conservation of species mass

33

å�� å{ + å��Ù�

åæÙ= − å

åæÙ(�B ,Ù� ) + Þ× , � = 1, … , )

Mass conservation

for species k

å�� å{ + å�(�Ù+B ,Ù)�

åæÙ= Þ× , � = 1, … , )

B ,Ù: the diffusion

velocity

å� ∑ � ��å{ + å��Ù ∑ � ��

åæÙ= − å

åæÙ� ∑B ,Ù� + > Þ×

�

�>

�

Gobla mass eq.å� ⋅ 1

å{ + å��¡ ⋅ 1åæ¡

= 0 + 0

∑B ,Ù� = 0 ∑Þ× = 0∑� = 1

Compute the diffusion velocity B An less accurate simple gradient model (Fick law )

34

B ,Ù� =−É é��é¨Õ

Fick law

In a simple condition when

we assume const É for all

species, i.e.

É0 = ⋯ = DØ … = D, B ,Ù� = É å�

åæÙîÓïðññ ò¥¡· = 0

>� violate: ∑B ,Ù� = 0

�

å�� å{ + å�(�Ù+îÓïðññ)�

åæÙ= − å

åæÙ(�B ,Ù� ) + Þ× , � = 1, … , )

îÓïðññ|¥¡· =∑É é��é¨Õ

Note: some CFD code does not

use this strategy of correction-

velocity, the inconsistence error

is then pumped into certain

abundant diluting gas such as N2

Compute the diffusion velocity B Solve the more accurate full equations

35

ó�Æ = ∑ ôõô�öõ�

B − BÆ� + �Æ − �Æ÷øù + ê

� ∑ �Æ� Æ̧ − ̧ , � for } = 1, . . )

ÉÆ = É Æ is binary mass diffusion of species } diffuse into �,

Neglect Soret effect (mass diffusion due to temperature gradient) .

mole

� = � �/� is the mole fraction of �,

Diffusion velocity B Binary diffusion in a two-species system �0 + � = 1 :

36

ó�Æ = ∑ ôõô�öõ�

B − BÆ� + �Æ − �Æ÷øù + ê

� ∑ �Æ� �̧ − ̧ , � for } = 1, . . )

Binary diffusion:

ó�0 = �0�É0

B0 − B

B0�0 = −É0ó�0

∑B � = B0 �0 + B� = 0

Fick law is exact for binary diffusion

Assume: |ó�| is mall, neglect volume force:

� = � �/�

Diffusion velocity B Multi-species diffusion: Hirschfelder-Curtiss approximation

37

Multi-species diffusion:

A complicated inversion problem, Hirschfelder-Curtiss approximation is a

best first-order approximation of exact system.

B � = −ÉØó�

É = 0È��∑ ôÕ/öÕ��Õú�

É ≠ ÉÙ species � diffuse

into the "mixture"

B � = −ÉØó�

not Fick law anymore

� = � �/�

ó�Æ = ∑ ôõô�öõ�

B − BÆ� + �Æ − �Æ÷øù + ê

� ∑ �Æ� �̧ − ̧ , � for } = 1, . . )

Species mass equations with different models of diffusion velocity

38

å�� å{ + å�(�Ù + BÙ·cdd|98 )�

åæÙ= − å

åæÙ(�É

� �

å� åæÙ

) + Þ× , � = 1, … , Ã

B � = −ÉØó� � = � �/�

B � = −ÉØ� � ó�

å� ∑ � ��å{ + å�(�Ù + BÙ·cdd|¥¡· ) ∑ � ��

åæÙ= − å

åæÙ� �É

å� åæÙ

+ > Þ× �

�

Fick approx. (not accurate, but easy for numerical implementation)

Hirschfelder-Curtiss

approx. (more accurate)

BÙ·cdd ò98 = ∑É � �

å� åæÙ

Various definition of Energy and enthalpy

39

Chemical energy: ∑ Δℎ!, " � @ E0 , ℎ!, " enthalpy of formation

Kinetic energy : 0 �¡�¡

Derive the kinetic energy equation from mass and momentum eq.s

40

å�å{ + å��Ù

åæÙ = 0

� å�¡å{ + ��Ù

å�¡åæÙ

= − å�åæ¡

+ åç¡ÙåæÙ

+ �∑� ̧,Ù Momentum eq.

û¡Ù = ç¡Ù − ��¡Ù

�¡ ×

� å 12 �¡

å{ + ��Ùå 1

2 �¡

åæÙ= �¡ − å�

åæ¡+ åç¡Ù

åæÙ+ �∑� ¸ ,Ù

éêʬ�³¬

é` + éê�Õʬ�³¬

é¨Õ≡ � öÊ

¬�³¬ö` = �¡(éü³Õ

é¨Õ + �∑� ̧,Ù)

12 �¡ ×

+

Useful indentiy: material-derivative � öýö` ≡ � éý

é` + �Ùé

é¨Õ­ = éêý

é` + éé¨Õ

��٭

viscous-stress contributes to “reversible” mechanical work!

Energy equation for total energy (sensible + chemisical-bond+ kinetic energy)

41

Total energy Z`� ÉZ`

É{ = − åÖÙ åæÙ

+ å åæÙ

û¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)�

\:× external

heat source

(not burning

released heat)

ÖÙ = −þ åDåæÙ

+ � > ℎ � B ,Ù�

Fourier’s

law Diffusion of multi-

species with

different enthalpy

û¡Ù = ç¡Ù − ��¡Ù

� ∑ � ̧,Ù(�Ù + B ,Ù)� , power produced by

volume force.

Buoyance, etc.

Useful indentiy: material-derivative � öýö` ≡ � éý

é` + �Ùé

é¨Õ­ = éêý

é` + éé¨Õ

��٭

Energy equationfor total enthalpy (sensible + chemistry+ kinetic energy)

42

Total Enthalpy: ℎ`=Z` + �/�

� ÉZ`É{ = − åÖÙ

åæÙ+ å

åæÙû¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)

�

� ÉZ`É{ = � Éℎ`

É{ − É�É{ − � å�¡

åæ¡

� Éℎ`É{ − É�

É{ − � å�¡åæ¡

= − åÖÙ åæÙ

+ å åæÙ

û¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)�

� Éℎ`É{ = å�

å{ − åÖÙ åæÙ

+ å åæÙ

ç¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)�

û¡Ù = ç¡Ù − ��¡Ù

Energy equationfor enthalpy (sensible + chemistry+ kinetic energy)

43

Enthalpy: ℎ=ℎ` − 0 �¡

� Éℎ`É{ = å�

å{ − åÖÙ åæÙ

+ å åæÙ

ç¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)�

� ÉℎÉ{ = É�

É{ − åÖÙ åæÙ

+ ç¡Ùå�¡åæÙ

+ \× + � > � ̧,ÙB ,Ù�

� É 12 �¡

É{ = �¡ − å�åæ¡

+ åç¡ÙåæÙ

+ �∑� ̧,¡

Energy equationfor sensible enthalpy (sensible + chemistry+ kinetic energy)

44

Sensible Enthalpy: ℎ� = ℎ − ∑ Δℎ!, " � @

� Éℎ�É{ = É�

É{ + ç¡Ùå�¡åæÙ

− åÖÙ åæÙ

+ ååæÙ

� > Δℎ!, " � B ,Ù�

− > Δℎ!, " Þ× + \× + � > � ̧,ÙB ,Ù

�

�

� ÉℎÉ{ = É�

É{ + ç¡Ùå�¡åæÙ

− åÖÙ åæÙ

+ \× + � > � ̧,ÙB ,Ù�

� É� É{ = − å

åæÙ�B ,Ù� + Þ× , � = 1, … , )> Δℎ!, " ×

�

ÖÙ = −þ åDåæÙ

+ � > ℎ � B ,Ù�

� Éℎ�É{ = É�

É{ + ç¡Ùå�¡åæÙ

+ å åæÙ

þ åDåæÙ

− å åæÙ

� > ℎ�, � B ,Ù�

− > Δℎ!, " Þ× + \× + � > � ̧,ÙB ,Ù

�

�

ℎ�, = ℎ − Δℎ!, "

Energy equation in temperature form

45

ℎ� ≡ � ��xDA�[�,`]

��

� Éℎ�É{ = ���

ÉDÉ{ + >��,Ò�

��Ò�á

�

Ò

ℎ�, ≡ � ��, xDA�(�,`)

��

� Éℎ�É{ = É�

É{ + ç¡Ùå�¡åæÙ

+ å åæÙ

þ åDåæÙ

− å åæÙ

� > ℎ�, � B ,Ù�

− > Δℎ!, " Þ×

�

+ \× + � > � ̧,ÙB ,Ù

�

�� ≡ > � (æ, {)��, �

� É� É{ = − å

åæÙ�B ,Ù� + Þ× >��,Ò ×

�

���ÉDÉ{ = É�

É{ + ç¡Ùå�¡åæÙ

+ å åæÙ

þ åDåæÙ

− � > ��, � B ,Ù�

å DåæÙ

− > ℎ Þ× + \× + � > � ̧,ÙB ,Ù�

�

ℎ = ℎ�, + Δℎ!, "

- -

Various form of energy eq.

46

Summary of reacting flow equationsassume no body force, no external heating

47

å�å{ + å��¡

åæ¡= 0

å��¡å{ + å��¡�Ù

åæÙ= − å�

åæ¡+ åç¡Ù

åæÙ

Global Mass

Momentum

å�� å{ + å��Ù�

åæÙ= − å

åæÙ�B ,Ù� + Þ× , � = 1, … , ) − 1Species

conservation

���ÉDÉ{ = É�

É{ + å åæÙ

þ åDåæÙ

− � > ��, � B ,Ù�

å DåæÙ

+ ç¡Ùå�¡åæÙ

− > Δℎ!, " Þ× �

Energy

� Éℎ`É{ = å�

å{ − åÖÙ åæÙ

+ å åæÙ

ç¡Ù�¡either

or

Simplification for the reacting flow governing equations

• Low Mach number assumption– �({,�) = �({) + �′({,�) and |pA| ≪ |�|

• “Thermodynamic” pressue + “hydrodynamic” pressure

• Transport coeff. ( such as Heat capacity )– Equal (among k) for all species

– Const (t) for mixture

• Non-dimentional number.– Lewis number (the ratio of thermal diffusivity to mass diffusivity. )

– Schmidt number (the ratio of momentum diffusivity (kinematic viscosity) and

mass diffusivity )

– Prandtl number (ratio of momentum diffusivity to thermal diffusivity)

48

Let’s consider a simple reacting system involving only two

species and a single step reaction

49

(e.g. 3 → 21 )Fuel → Product

Mass fraction of:

Product: �Fuel : 1 − �

Assumption:

(1) 1D

(2) Equal molecular weight: �!�^r = ��dc© = � → �� = ��)

(3) Δℎ!�^r" = 0, Δℎ�dc©" < 0 (heat release, exothermal reaction)

(4) Constant thermodynamic/transport properties for fuel/product and perfect ideal gas,

heat capacity: ��, ��mass diffusivity: É! = É� = É0 (to be used later)

(more)

Z` = ��D + �2 + Δℎ!,�" �

ℎ` = Z` + ��

p = �C∗D; �� − �� = C∗

Obtain the reduced equations for a simplified reacting flow system

(1) the species-mass equation

50

å�� å{ + å�(�Ù+îÓïðññ)�

åæÙ= − å

åæÙ�B ,Ù� + Þ× , � = 1, … , )

å��å{ + å���

åæ = − ååæ ÉA å

åæ � + Þ× �dc© , � = 1,2

3D�1D Fick law

Only two speciesÉA ≡ �É

Assume const.

Obtain the reduced equations for a simplified reacting flow system

(2) the momentum equation:

51

å��å{ + å(�� + �)

åæ = ååæ è′ å

åæ �

3D�1D

Assume const.

å��¡å{ + å��¡�Ù

åæÙ= − å�

åæ¡+ åç¡Ù

åæÙ+ � > � ̧,Ù

@

E0

ç¡Ù = è(é�³é¨Õ

+é�Õ騳

) − 1 �¡Ù

��

è′ ≡ 43 è

Obtain the reduced equations for a simplified reacting flow system

(3) energy equation:

52

åå{ �Z` + å

åæ ��ℎ` = ååæ þ å

åæ D

Assume þ const.

� Éℎ`É{ = å�

å{ − åÖÙ åæÙ

+ å åæÙ

ç¡Ù�¡ + \× + � > � ̧,Ù(�Ù + B ,Ù)�

Neglect viscous heating

���ÉDÉ{ = É�

É{ + å åæÙ

þ åDåæÙ

− � > ��, � B ,Ù�

å DåæÙ

+ ç¡Ùå�¡åæÙ

− > Δℎ!, " Þ× �

���ÉDÉ{ = å�

å{ + å åæÙ

þ åDåæÙ

− Δℎ!,�dc©" Þ× �dc©

∑B ,Ù� = 0

Compressible (Conservative form)

Low Mach number assumption:

�(æ, {) = �({) + �A(æ, {), |�A | ≪ |�|Non-conservative form:

The simplified 1D reacting systemSummary for the compressible reacting flow governing equations

53

Fuel → Product Mass fraction of:

Product: �Fuel : 1 − �

Þ× �dc© = 1ç·

(1 − �) ZÈ�Ð�å��å{ + å���

åæ = É′ å�åæ + Þ× �dc©

å�å{ + å��

åæ = 0åå{ �� + å

åæ �� + � = èA å�

åæåå{ �Z` + å

åæ ��ℎ` = þ åDåæ

Conservation laws:

Specie mass:“product”

Total mass:

Momentum:

Energy:

Arrhenius reaction

Z` = ��D + �2 + Δℎ!,�" �

ℎ` = Z` + ��

�� − �� = C∗

Equation of state � = �C∗D(Note: if diffusion, viscous and heat-conduction terms are neglected, the system is

governed by a hyperbolic four-waves equations, all equations are in conservative

form except an non-zero source term in the first species-mass equation)

The simplified 1D reacting systemSummary of governing equations under low Mach assumption

54

Fuel → Product Mass fraction of:

Product: �Fuel : 1 − �

Þ× �dc© = 1ç·

(1 − �) ZÈ�Ð�å��å{ + å���

åæ = É′ å�åæ + Þ× �dc©

å�å{ + å��

åæ = 0åå{ �� + å

åæ �� + �′ = è′ å�åæ

Conservation law for:

Specie mass:“product”

total mass:

Momentum:

Energy:

Arrhenius reaction

���ÉDÉ{ = å�

å{ + å åæÙ

þ åDåæÙ

− Δℎ!,�dc©" Þ× �dc©

Low Mach assumption:

� { = � {, æ C∗D({, æ) , �A(æ, {) ≠ �({)