اﺣﺘﺮاق و ﺗﺮﻣﻮﺷﯿﻤﯽ

Advanced Numerical Methods 1

ﯽﻤﯿﺷﻮﻣﺮﺗ و قاﺮﺘﺣا

Advanced Combustion

Advanced Numerical Methods 2

ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ )

Chemical Equilibrium (

رد 

يﺎﻫﺪﻨﯾآﺮﻓ ﯽﻗاﺮﺘﺣا

ﺎﻣد ﻻﺎﺑ ) T>1250 K (،

تﻻﻮﺼﺤﻣ قاﺮﺘﺣا

ﺮﮕﯾد طﻮﻠﺨﻣ

يا هدﺎﺳ

زا .ﺪﻨﺘﺴﯿﻧ .(ﺪﻨﮐ ﯽﻣ ﻦﯿﯿﻌﺗ يﺮﺘﻣﻮﯿﮐﻮﺘﺳا ﺶﻨﮐاو ﻪﮐ ﯽﺗﻻﻮﺼﺤﻣ) لآ هﺪﯾا تﻻﻮﺼﺤﻣ رد 

ﻦﯾا ﺖﻟﺎﺣ يﺎﻫ ﻪﻧﻮﮔ

ﯽﻠﺻا ﻪﯾﺰﺠﺗ

هﺪﺷ )

dissociate (

و ﯽﻫوﺮﮔ زا

يﺎﻫ ﻪﻧﻮﮔ

ﯽﻋﺮﻓ .ﺪﻧﺮﯿﮔ ﯽﻣ ﻞﮑﺷ

تﻻﻮﺼﺤﻣ



قاﺮﺘﺣا ﻞﻣﺎﮐ

) نوﺪﺑ ﻪﯾﺰﺠﺗ

ﯽﯾﺎﯿﻤﯿﺷ (:

<1



ϕ

=1



ϕ

<1



ϕ

تﻻﻮﺼﺤﻣ



قاﺮﺘﺣا ﺎﺑ

ﻪﯾﺰﺠﺗ ﯽﯾﺎﯿﻤﯿﺷ

:

Hydrocarbon+AirCO

₂

+H

₂

O+O

₂

+N

₂

Hydrocarbon+AirCO

₂

+CO+H

₂

O+N

₂

Hydrocarbon+AirCO

₂

+H

₂

O+N

₂

Advanced Combustion

Hydrocarbon+AirCO

₂

+CO+H

₂

O+O

₂

+O+OH+H+N

₂

+NO+N…

Advanced Numerical Methods 3

ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ )

Chemical Equilibrium (

ﺮﮔا



ﭻﯿﻫ يﺮﯿﯿﻐﺗ

رد ﺐﯿﮐﺮﺗ ﯽﯾﺎﯿﻤﯿﺷ

ﮏﯾ ﻢﺘﺴﯿﺳ

ﻪﺑ دﻮﺟو

،ﺪﯾﺎﯿﻧ نآ

ﯿﺳ ﻢﺘﺴ

رد . دﺮﺑ ﯽﻣ ﺮﺳ ﻪﺑ ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ

مﻮﻬﻔﻣ



لدﺎﻌﺗ ﯽﺋﺎﯿﻤﯿﺷ

ﻪﺸﯾر رد

نﻮﻧﺎﻗ مود

ﮏﯿﻣﺎﻨﯾدﻮﻣﺮﺗ دراد

.

لدﺎﻌﺗ



ﯽﯾﺎﯿﻤﯿﺷ ياﺮﺑ

ﮏﯾ ﻢﺘﺴﯿﺳ

ﻪﺘﺴﺑ

ﺶﻨﮐاو ﺮﯾز

ار رد ﺮﻈﻧ ﺪﯾﺮﯿﮕﺑ :

ﺮﮔا يﺎﻣد ﯽﯾﺎﻬﻧ

ﻪﺑ هزاﺪﻧا ﯽﻓﺎﮐ

دﺎﯾز ﺪﺷﺎﺑ

يد ﺪﯿﺴﮐا ﻦﺑﺮﮐ

ﻪﯾﺰﺠﺗ ﺪﻫاﻮﺧ

ﺪﺷ .

يﺮﺴﮐ α زا

يد ﺪﯿﺴﮐا ﻦﺑﺮﮐ

ﺖﺳا ﻪﮐ

ﻪﯾﺰﺠﺗ هﺪﺷ

ﺖﺳا .

ناﻮﺗ ﯽﻣ يﺎﻣد

ﻪﻠﻌﺷ ﮏﯿﺗﺎﺑﺎﯾدآ

ار ياﺮﺑ ﺶﻨﮐاو

ﻪﯾﺰﺠﺗ قﻮﻓ

بﺎﺴﺣ دﺮﮐ

. ﻦﯾا ﺎﻣد ﯽﻌﺑﺎﺗ زا

α .ﺖﺳا

Advanced Combustion

CO+0.5O₂CO₂

CO+0.5O₂ (1-α)CO₂+ αCO+0.5αO₂

Advanced Numerical Methods 4

لدﺎﻌﺗ ﻪﻟوﺰﯾا ﻪﺘﺴﺑ ﻢﺘﺴﯿﺳ ﮏﯾ ياﺮﺑ ﯽﯾﺎﯿﻤﯿﺷ

Advanced Combustion

Fraction of CO₂ un-dissociated (1-α)

dS ≥ 0

ﮐ ﯽﻣ اﺪﯿﭘ ﺖﻔﯿﺷ ﻢﻤﯾﺰﮐﺎﻣ ﯽﭘوﺮﺘﻧآ ﺎﺑ ﻪﻄﻘﻧ ﺖﻤﺳ ﻪﺑ دﻮﺧ ﻪﺑ دﻮﺧ ﻢﺘﺴﯿﺳ ﺐﯿﮐﺮﺗ 

ﺪﻨ . نﺪﯿﺳر ﺎﺑ

ﺪﻫد يور ﺪﻧاﻮﺗ ﯽﻤﻧ ﻢﺘﺴﯿﺳ ﺐﯿﮐﺮﺗ رد يﺮﯿﯿﻐﺗ ﭻﯿﻫ ﯽﭘوﺮﺘﻧآ ﻢﻤﯾﺰﮐﺎﻣ ﻪﻄﻘﻧ ﻪﺑ .

(dS)

_U,V,m

=0

ﺪﻨﯾآﺮﻓ مﺎﺠﻧا ﺖﻬﺟ 

:

لدﺎﻌﺗ طﺮﺷ 

:

Advanced Numerical Methods 5

ﮏﯿﻣﺎﻨﯾدﻮﻣﺮﺗ مود نﻮﻧﺎﻗ :

ﺲﺒﯿﮔ دازآ يژﺮﻧا

يژﺮﻧا



دازآ ﺲﺒﯿﮔ

) Gibbs (

ﺎﺑ



ﯽﻓﺮﻌﻣ يژﺮﻧا

دازآ ﺲﺒﯿﮔ

:

𝒅𝒅𝑮𝑮 = 𝒅𝒅𝒅𝒅 + 𝑷𝑷𝒅𝒅𝑷𝑷 + 𝑷𝑷𝒅𝒅𝑷𝑷 − 𝑻𝑻𝒅𝒅𝑻𝑻 − 𝑻𝑻𝒅𝒅𝑻𝑻 dP=0

dT=0

ﻢﯾراد رﺎﮐ و ﺮﺳ ﺖﺑﺎﺛ رﺎﺸﻓ و ﺎﻣد ﻂﯾاﺮﺷ ﺎﺑ ﯽﻗاﺮﺘﺣا يﺎﻫدﺮﺑرﺎﮐ زا يرﺎﯿﺴﺑ رد :

Advanced Combustion

𝟏𝟏^{𝒔𝒔𝒔𝒔} 𝑳𝑳𝑳𝑳𝑳𝑳: 𝒅𝒅𝒅𝒅 = 𝜹𝜹𝜹𝜹 − 𝜹𝜹𝑾𝑾 = 𝜹𝜹𝜹𝜹 − 𝐏𝐏𝐏𝐏𝐏𝐏 𝟐𝟐^{𝒏𝒏𝒅𝒅} 𝑳𝑳𝑳𝑳𝑳𝑳: 𝒅𝒅𝑻𝑻 ≥ 𝜹𝜹𝜹𝜹

𝑻𝑻

𝒅𝒅𝒅𝒅 + 𝐏𝐏𝐏𝐏𝐏𝐏 − 𝐓𝐓𝐏𝐏𝐓𝐓 ≤ 𝟎𝟎

𝑮𝑮 = 𝑯𝑯 − 𝑻𝑻𝑻𝑻 = 𝒅𝒅 + 𝑷𝑷𝑷𝑷 − 𝑻𝑻𝑻𝑻

𝒅𝒅𝑮𝑮 = 𝒅𝒅𝒅𝒅 + 𝑷𝑷𝒅𝒅𝑷𝑷 − 𝑻𝑻𝒅𝒅𝑻𝑻 ≤ 𝟎𝟎

ياﺮﺑ 

ﮏﯾ ﻢﺘﺴﯿﺳ ﻪﺘﺴﺑ

ﺎﺑ ﺎﻣد و رﺎﺸﻓ ﺖﺑﺎﺛ

ﺶﻨﮐاو ﯽﯾﺎﯿﻤﯿﺷ

رد ﺖﻬﺟ ﺶﻫﺎﮐ

يژﺮﻧا آ

داز .ﺪﻨﮐ ﯽﻣ ﺖﮐﺮﺣ ﺲﺒﯿﮔ

Advanced Numerical Methods 6

ﮏﯿﻣﺎﻨﯾدﻮﻣﺮﺗ مود نﻮﻧﺎﻗ :

ﺲﺒﯿﮔ دازآ يژﺮﻧا

Advanced Combustion

𝒅𝒅𝑮𝑮 _{𝑻𝑻,𝑷𝑷} < 𝟎𝟎 𝒅𝒅𝑮𝑮 _{𝑻𝑻,𝑷𝑷,𝒎𝒎} = 𝟎𝟎

ﺪﻨﯾآﺮﻓ مﺎﺠﻧا ﺖﻬﺟ 

:

لدﺎﻌﺗ طﺮﺷ 

:

Advanced Numerical Methods 7

ﮏﯿﻣﺎﻨﯾدﻮﻣﺮﺗ مود نﻮﻧﺎﻗ :

ﺲﺒﯿﮔ دازآ يژﺮﻧا

Advanced Combustion

ياﺮﺑ 

ﯽﻃﻮﻠﺨﻣ زا

يﺎﻫزﺎﮔ

،لآ هﺪﯾا ﻊﺑﺎﺗ

ﺲﺒﯿﮔ ﻪﻧﻮﮔ

ما i زا ﻪﻄﺑار ﺮﯾز

ﺖﺳد ﻪﺑ ﺪﯾآ ﯽﻣ

:

ﻪﺑ ﻪﮐ يرﻮﻃ

�𝒈𝒈_{𝒊𝒊,𝑻𝑻}^𝟎𝟎 ﻊﺑﺎﺗ

ﺲﺒﯿﮔ ﺮﺼﻨﻋ

ﺺﻟﺎﺧ رد

رﺎﺸﻓ دراﺪﻧﺎﺘﺳا

و 𝑷𝑷_𝒊𝒊 رﺎﺸﻓ ﯽﺋﺰﺟ

ﻪﻧﻮﮔ ﺖﺳا

.

ياﺮﺑ



ﻢﺘﺴﯿﺳ ﯽﺸﻨﮐاو

ﻊﺑﺎﺗ ﺲﺒﯿﮔ

ﻞﯿﮑﺸﺗ ترﻮﺻ ﻪﺑ

ﺮﯾز ﻒﯾﺮﻌﺗ ﻮﺷ ﯽﻣ

د:

ﺎﻫ v’

ﺐﯾﺮﺿ يﺮﺘﻣﻮﯿﮐﻮﺘﺳا

يﺎﻫ نﺎﻤﻟا ﻞﯿﮑﺸﺗ

هﺪﻨﻫد ﻪﻧﻮﮔ

درﻮﻣ ﺮﻈﻧ

ﺪﻨﺘﺴﻫ .

ﻊﺑﺎﺗ



ﺲﺒﯿﮔ ياﺮﺑ

ﯽﻃﻮﻠﺨﻣ زا

يﺎﻫزﺎﮔ لآ هﺪﯾا

ترﻮﺻ ﻪﺑ ﺮﯾز

ﻒﯾﺮﻌﺗ دﻮﺷ ﯽﻣ

:

�𝒈𝒈_{𝒊𝒊,𝑻𝑻} = �𝒈𝒈_{𝒊𝒊,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊 𝑷𝑷_𝟎𝟎

�𝒈𝒈_{𝒇𝒇,𝒊𝒊}^𝟎𝟎 𝑻𝑻 = �𝒈𝒈_{𝒊𝒊,𝑻𝑻}^𝟎𝟎 − �

𝒋𝒋 𝒆𝒆𝑻𝑻𝒆𝒆𝒎𝒎𝒆𝒆𝒏𝒏𝒔𝒔𝒔𝒔

́𝒗𝒗_𝒋𝒋 �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎

𝑮𝑮_{𝒎𝒎𝒊𝒊𝒎𝒎} = � 𝑵𝑵_𝒊𝒊 �𝒈𝒈_{𝒊𝒊,𝑻𝑻} = � 𝑵𝑵_𝒊𝒊 �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊 𝑷𝑷_𝟎𝟎

Advanced Numerical Methods 8

ﮏﯿﻣﺎﻨﯾدﻮﻣﺮﺗ مود نﻮﻧﺎﻗ :

ﺲﺒﯿﮔ دازآ يژﺮﻧا

Advanced Combustion

ياﺮﺑ 

ﺎﻣد و رﺎﺸﻓ ﺖﺑﺎﺛ

طﺮﺷ لدﺎﻌﺗ

ﺖﺳد ﻪﺑ ﺪﻣآ

:

زا 

ﯽﯾﺎﺠﻧآ ﻪﮐ

ﺎﻣد و رﺎﺸﻓ طﻮﻠﺨﻣ

ﺖﺑﺎﺛ ﺖﺳا

.)

∑ 𝒅𝒅𝑷𝑷_𝒊𝒊 = 𝟎𝟎 (

𝒅𝒅𝑮𝑮

_{𝒎𝒎𝒊𝒊𝒎𝒎}

= 𝟎𝟎

� 𝒅𝒅𝑵𝑵_𝒊𝒊 �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊

𝑷𝑷_𝟎𝟎 + � 𝑵𝑵_𝒊𝒊 𝒅𝒅 �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊

𝑷𝑷_𝟎𝟎 = 𝟎𝟎

𝒅𝒅𝑮𝑮_{𝒎𝒎𝒊𝒊𝒎𝒎} = 𝟎𝟎 = � 𝒅𝒅𝑵𝑵_𝒊𝒊 �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊 𝑷𝑷_𝟎𝟎

Advanced Numerical Methods 9

ﺺﺨﺸﻣ رﺎﺸﻓ و ﺎﻣد رد هدﺎﺳ ﺶﻨﮐاو ﮏﯾ ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ

Advanced Combustion

ﺶﻨﮐاو



ﺮﯾز ﻪﮐ رد ﺎﻣد و رﺎﺸﻓ ﺖﺑﺎﺛ

يور ﺪﻫد ﯽﻣ

ار رد ﺮﻈﻧ ﺪﯾﺮﯿﮕﺑ

:

ضﺮﻓ 

ﻢﯿﻨﮐ ﯽﻣ ﻪﮐ

4 ﻪﻧﻮﮔ قﻮﻓ

رد لﺎﺣ لدﺎﻌﺗ

ﯽﯾﺎﯿﻤﯿﺷ ﺪﻨﺘﺴﻫ

. ﺲﭙﺳ ﺶﻨﮐاو

ﻪﺑ هزاﺪﻧا

ﺖﯾﺎﻬﻨﯿﺑ ﮏﭼﻮﮐ

رد ﺖﻬﺟ ﺖﻓر

ﺶﯿﭘ يور

ﺪﻨﮐ ﯽﻣ .

ﻦﯾا عﻮﺿﻮﻣ ﺚﻋﺎﺑ

ﺶﻫﺎﮐ داﺪﻌﺗ

لﻮﻣ .دﻮﺷ ﯽﻣ ^D و ^C لﻮﻣ ﺶﯾاﺰﻓا و ^B و ^A يﺎﻫ ﻪﻧﻮﮔ

ﺮﯿﯿﻐﺗ 

رد داﺪﻌﺗ لﻮﻣ

ﺮﻫ ﻪﻧﻮﮔ ﯽﯾﺎﯿﻤﯿﺷ

ﺐﺳﺎﻨﺘﻣ ﺎﺑ

ﺐﯾﺮﺿ يﺮﺘﻣﻮﯿﮐﻮﺘﺳا

نآ ﺑ ﯽﻣ ﺪﺷﺎ :

راﺪﻘﻣ 

ﻒﻟﺎﺨﻣ 𝜺𝜺 ﺮﻔﺻ

ﺖﺳا .

ﻦﯿﻨﭽﻤﻫ P₀=1 atm

.

𝒗𝒗

_𝑨𝑨

𝑨𝑨 + 𝒗𝒗

_𝑩𝑩

𝑩𝑩 ⇔ 𝒗𝒗

_𝒄𝒄

𝑪𝑪 + 𝒗𝒗

_𝒅𝒅

𝑫𝑫

ε ν

ε ν ε

ν ε

ν ε

ν

i i

D D

C C

B B

A A

dn

dn dn

dn dn

=

⇒

=

=

−

=

−

=

𝒅𝒅𝑮𝑮 _{𝑻𝑻,𝑷𝑷} = � �𝒈𝒈_{𝒋𝒋,𝑻𝑻}^𝟎𝟎 + 𝑹𝑹_𝒖𝒖𝑻𝑻𝑻𝑻𝒏𝒏 𝑷𝑷_𝒊𝒊

𝑷𝑷_𝟎𝟎 𝒗𝒗_𝒊𝒊𝜺𝜺 = 𝟎𝟎 لدﺎﻌﺗ طﺮﺷ

Advanced Numerical Methods 10

ﺺﺨﺸﻣ رﺎﺸﻓ و ﺎﻣد رد هدﺎﺳ ﺶﻨﮐاو ﮏﯾ ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ

Advanced Combustion

−�𝒈𝒈_𝑨𝑨^𝟎𝟎𝒗𝒗_𝑨𝑨 − �𝒈𝒈_𝑩𝑩^𝟎𝟎𝒗𝒗_𝑩𝑩 + �𝒈𝒈_𝑪𝑪^𝟎𝟎𝒗𝒗_𝑪𝑪 + �𝒈𝒈_𝑫𝑫^𝟎𝟎 𝒗𝒗_𝑫𝑫

+ 𝑹𝑹_𝒖𝒖𝑻𝑻 −𝒗𝒗_𝑨𝑨𝑻𝑻𝒏𝒏𝑷𝑷_𝑨𝑨 − 𝒗𝒗_𝑩𝑩𝑻𝑻𝒏𝒏𝑷𝑷_𝑩𝑩 + 𝒗𝒗_𝑪𝑪𝑻𝑻𝒏𝒏𝑷𝑷_𝑪𝑪 + 𝒗𝒗_𝑫𝑫𝑻𝑻𝒏𝒏𝑷𝑷_𝑫𝑫 = 𝟎𝟎

ﻒﯾﺮﻌﺗ ﺎﺑ

∆𝑮𝑮^𝟎𝟎 = −�𝒈𝒈_𝑨𝑨^𝟎𝟎𝒗𝒗_𝑨𝑨 − �𝒈𝒈_𝑩𝑩^𝟎𝟎𝒗𝒗_𝑩𝑩 + �𝒈𝒈_𝑪𝑪^𝟎𝟎𝒗𝒗_𝑪𝑪 + �𝒈𝒈_𝑫𝑫^𝟎𝟎 𝒗𝒗_𝑫𝑫 :

ﺖﺷاد ﻢﯿﻫاﻮﺧ

−∆𝑮𝑮^𝟎𝟎 :

𝑹𝑹_𝒖𝒖𝑻𝑻 = 𝐥𝐥𝐥𝐥 𝑷𝑷_𝑪𝑪^{𝒗𝒗𝑪𝑪} 𝑷𝑷_𝑫𝑫^{𝒗𝒗𝑫𝑫} 𝑷𝑷_𝑨𝑨^{𝒗𝒗𝑨𝑨} 𝑷𝑷_𝑩𝑩^{𝒗𝒗𝑩𝑩}

،ﯽﮔدﺎﺳ ياﺮﺑ لدﺎﻌﺗ ﺖﺑﺎﺛ

K_P دﻮﺷ ﯽﻣ ﻒﯾﺮﻌﺗ ﺮﯾز ترﻮﺻ ﻪﺑ ،

−∆𝑮𝑮^𝟎𝟎 :

𝑹𝑹_𝒖𝒖𝑻𝑻 = 𝐥𝐥𝐥𝐥𝑲𝑲_𝑷𝑷 𝐊𝐊_𝐏𝐏 = 𝑷𝑷_𝑪𝑪^{𝒗𝒗𝑪𝑪} 𝑷𝑷_𝑫𝑫^{𝒗𝒗𝑫𝑫} 𝑷𝑷_𝑨𝑨^{𝒗𝒗𝑨𝑨} 𝑷𝑷_𝑩𝑩^{𝒗𝒗𝑩𝑩}

total i

i X P

P =

B A D C B

A C D

total B

A

D C

P P

X X

X T X

K _{ν ν} ^{ν ν ν ν}

ν

ν + − −

= ) (

Advanced Numerical Methods 11 Advanced Combustion

لدﺎﻌﺗ ﺖﺑﺎﺛ

Logarithms to base e of the equilibrium constant K_p

The equilibrium constant K_p for the reaction v_AA + v_BB  v_CC + v_DD is defined as:

B A

D C

v B v A

v D v C

p P P

P K ≡ P

Temp.

K H₂2H O₂ 2O N₂ 2N H₂O H₂

+½O₂ H₂O ½H₂

+OH CO₂ CO

+½O₂ ½N₂+½O₂ NO 298500

10001200 14001600 18002000 22002400 26002800 30003200 34003600 38004000 45005000 55006000

-164.005 -92.827 -39.803 -30.874 -24.463 -19.637 -15.866 -12.840 -10.353 -8.276 -6.517 -5.002 -3.685 -2.534 -1.516 -0.609 0.202 0.934 2.486 3.725 4.743 5.590

-186.975 -105.630 -45.150 -35.005 -27.742 -22.285 -18.030 -14.622 -11.827 -9.497 -7.521 -5.826 -4.357 -3.072 -1.935 -0.926 -0.019 0.796 2.513 3.895 5.023 5.963

-367.480 -213.372 -99.127 -80.011 -66.329 -56.055 -48.051 -41.645 -36.391 -32.011 -28.304 -25.117 -22.359 -19.937 -17.800 -15.898 -14.199 -12.660 -9.414 -6.807 -4.666 -2.865

-92.208 -52.691 -23.163 -18.182 -14.609 -11.921 -9.826 -8.145 -6.768 -5.619 -4.648 -3.812 -3.086 -2.451 -1.891 -1.392 -0.945 -0.542 0.312 0.996 1.560 2.032

-106.208 -60.281 -26.034 -20.283 -16.099 -13.066 -10.657 -8.728 -7.148 -5.832 -4.719 -3.763 -2.937 -2.212 -1.576 -1.088 -0.501 -0.044 0.920 1.689 2.318 2.843

-103.762 -57.616 -23.529 -17.871 -13.842 -10.830 -8.497 -6.635 -5.120 -3.860 -2.801 -1.894 -1.111 -0.429 0.169 0.701 1.176 1.599 2.490 3.197 3.771 4.245

-35.052 -20.295 -9.388 -7.569 -6.270 -5.294 -4.536 -3.931 -3.433 -3.019 -2.671 -2.372 -2.114 -1.888 -1.690 -1.513 -1.356 -1.216 -0.921 -0.686 -0.497 -0.341

Advanced Numerical Methods 12 Advanced Combustion

لدﺎﻌﺗ ﺖﺑﺎﺛ

Logarithms to base 10 of the equilibrium constant K_p

Advanced Numerical Methods 13

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



9:

ﺐﯿﮐﺮﺗ ﯽﻟدﺎﻌﺗ

ﻪﯾﺰﺠﺗ ﮏﯾ

لﻮﻣ CO₂

ﻪﺑ و CO

O₂ رد يﺎﻣد 3200 K

و رﺎﺸﻓ 1 atm

ار ﻦﯿﯿﻌﺗ

ﺪﯿﻨﮐ .

ﺶﻨﮐاو ﯽﻟدﺎﻌﺗ

ﻪﯾﺰﺠﺗ CO₂

ترﻮﺻ ﻪﺑ ﺮﯾز

ﺖﺳا :

ﻦﯾا ﺶﻨﮐاو ار

ناﻮﺗ ﯽﻣ ﻪﺑ

ترﻮﺻ ﺮﯾز

ﯽﺴﯾﻮﻧزﺎﺑ دﺮﮐ

.

ﻦﯾاﺮﺑﺎﻨﺑ :

CO₂ CO+0.5O₂ v₁CO₂ v₃CO+v₄O₂

𝐊𝐊_𝐏𝐏 = 𝑿𝑿_{𝑪𝑪𝑪𝑪}^{𝒗𝒗𝟑𝟑} 𝑿𝑿_𝑪𝑪^{𝒗𝒗𝟒𝟒}_𝟐𝟐

𝑿𝑿_{𝑪𝑪𝑪𝑪}^{𝒗𝒗𝟏𝟏} _𝟐𝟐 𝑷𝑷_{𝒔𝒔𝒕𝒕𝒔𝒔}^{𝒗𝒗𝟑𝟑+𝒗𝒗𝟒𝟒−𝒗𝒗𝟏𝟏} 𝐊𝐊_𝐏𝐏 = 𝑿𝑿_{𝑪𝑪𝑪𝑪} 𝑿𝑿_𝑪𝑪^{𝟏𝟏 𝟐𝟐⁄}_𝟐𝟐

𝑿𝑿_{𝑪𝑪𝑪𝑪𝟐𝟐} 𝟏𝟏 ^{𝟏𝟏 𝟐𝟐⁄}

Advanced Numerical Methods 14

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



9 ) ﻪﻣادا (:

رد ﻪﮐ ﯽﺗرﻮﺻ رد

ﺶﻨﮐاو ﯽﻌﻗاو

CO₂ ﺎﺗ

ﻪﺟرد

Advanced Numerical Methods 15

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



10 :

ﯽﯾﺎﻣد ﻪﮐ

رد نآ 5

% نﮋﯿﺴﮐا ود

ﯽﻤﺗا )

O₂ ( رد رﺎﺸﻓ 3 atm

ﻪﺑ نﮋﯿﺴﮐا ﮏﺗ

ﯽﻤﺗا )

(O ﻪﯾﺰﺠﺗ

دﻮﺷ ﯽﻣ ار

ﻦﯿﯿﻌﺗ ﺪﯿﻨﮐ

.

ياﺮﺑ ﺖﻟﻮﻬﺳ

1 kmol زا

O₂ ار رد ﺮﻈﻧ ﻢﯾﺮﯿﮔ ﯽﻣ .

ﺶﻨﮐاو ﻪﯾﺰﺠﺗ

ﯽﻟدﺎﻌﺗ O₂

ﻪﺑ ﻞﮑﺷ ﺮﯾز

ﺖﺳا :

رد ﻪﮐ ﯽﺗرﻮﺻ رد

ﺶﻨﮐاو ﯽﻌﻗاو

O₂ ﺎﺗ ﻪﺟرد

Advanced Numerical Methods 16

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



10 ) ﻪﻣادا (:

ﺎﺑ عﻮﺟر ﻪﺑ

لوﺪﺟ طﻮﺑﺮﻣ

ﻪﺑ يﺎﻫ ﺖﺑﺎﺛ لدﺎﻌﺗ

يﺎﻣد ﻢﺘﺴﯿﺳ

ﺖﺳد ﻪﺑ ﺪﯾآ ﯽﻣ

:

يﺎﻣد ﯽﯾﺎﻬﻧ

ﺎﺑ ﯽﺑﺎﯾ نﺎﯿﻣ ﻦﯿﺑ

ود يﺎﻣد و 3100

ﻦﯾﻮﻠﮐ 3200 ﺖﺳد ﻪﺑ

ﺪﯾآ ﯽﻣ .

𝑻𝑻𝒕𝒕𝒈𝒈𝐊𝐊_𝐏𝐏 = 𝑻𝑻𝒕𝒕𝒈𝒈 𝟎𝟎.𝟎𝟎𝟑𝟑𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = −𝟏𝟏. 𝟎𝟎𝟐𝟐𝟐𝟐

𝑻𝑻 = 𝟑𝟑𝟏𝟏𝟑𝟑𝟐𝟐.𝟔𝟔 𝐊𝐊

ﻪﯾﺰﺠﺗ ﺪﺻرد دوﺮﺑ ﺮﺗﻻﺎﺑ ﺎﻣد ﺮﮔا  O₂

دﻮﺷ ﯽﻣ ﺮﺘﺸﯿﺑ .

Advanced Numerical Methods 17

لدﺎﻌﺗ ﺖﺑﺎﺛ درﻮﻣ رد ﯽﺗﺎﮑﻧ

Advanced Combustion

ﺖﺑﺎﺛ



لدﺎﻌﺗ

K_P

ﺎﻬﻨﺗ ﯽﻌﺑﺎﺗ

زا ﺎﻣد ﺖﺳا .

ﺖﺑﺎﺛ 

لدﺎﻌﺗ ﺶﻨﮐاو

سﻮﮑﻌﻣ 1/K_P

ﺖﺳا .

ﺮﻫ 

ﻪﭼ K_P ﺮﺗ گرﺰﺑ

،ﺪﺷﺎﺑ قاﺮﺘﺣا

ﺮﺗ ﻞﻣﺎﮐ دﻮﺷ ﯽﻣ

.

دﻮﺟو 

زﺎﮔ ﯽﺑ ﺮﺛا ﺮﺑ يور ﺐﯿﮐﺮﺗ

ﯽﻟدﺎﻌﺗ ﺮﺛا

ﯽﻣ دراﺬﮔ ﻪﻧ

ﺮﺑ يور K_P

.

ﺮﻫ 

هﺎﮔ ﺐﺋاﺮﺿ يﺮﺘﻣﻮﯿﮐﻮﺘﺳا

2 ﺮﺑاﺮﺑ

،دﻮﺷ راﺪﻘﻣ

K_P ﻪﺑ ناﻮﺗ ود

ﺪﺳر ﯽﻣ .

H₂+0.5O₂ H₂O at 1 atm H₂+0.5O₂ H₂O at 5 atm H₂+0.5O₂+3N₂ H₂O+3N₂ at 3 atm H₂+2O₂+5N₂ H₂O+1.5O₂+5N₂ at 2 atm

H₂+0.5O₂ H₂O

H₂OH₂+0.5O₂

𝐊𝐊_𝐏𝐏 = 𝑿𝑿_𝑯𝑯^𝟏𝟏_{𝟐𝟐𝑪𝑪}

𝑿𝑿_𝑯𝑯^𝟏𝟏_𝟐𝟐 𝑿𝑿_𝑪𝑪^{𝟎𝟎.𝟎𝟎}_𝟐𝟐 𝑷𝑷_{𝒔𝒔𝒕𝒕𝒔𝒔}^{−𝟎𝟎.𝟎𝟎} 𝐊𝐊_𝐏𝐏 = 𝑿𝑿_𝑯𝑯^𝟏𝟏_𝟐𝟐 𝑿𝑿_𝑪𝑪^{𝟎𝟎.𝟎𝟎}_𝟐𝟐

𝑿𝑿_𝑯𝑯^𝟏𝟏_{𝟐𝟐𝑪𝑪} 𝑷𝑷_{𝒔𝒔𝒕𝒕𝒔𝒔}^{+𝟎𝟎.𝟎𝟎}

Advanced Numerical Methods 18

ﺺﺨﺸﻣ رﺎﺸﻓ و ﺎﻣد رد نﺎﻣﺰﻤﻫ يﺎﻫ ﺶﻨﮐاو ﯽﯾﺎﯿﻤﯿﺷ لدﺎﻌﺗ

Advanced Combustion

ضﺮﻓ



ﺪﯿﻨﮐ ود

ﺶﻨﮐاو ﺮﯾز

ترﻮﺻ ﻪﺑ نﺎﻣﺰﻤﻫ

رد ﺎﻣد و رﺎﺸﻓ ﺖﺑﺎﺛ

قﺎﻔﺗا ﯽﻣ

ﺘﻓا ﺪ :

ﺎﺑ 

لﺎﻤﻋا طﺮﺷ

لدﺎﻌﺗ ياﺮﺑ

ﻦﯾا ود ﺶﻨﮐاو نﺎﻣﺰﻤﻫ

و ﺎﺑ يﺪﻧور ﻪﺑﺎﺸﻣ

ﺎﺑ تﺎﺒﺳﺎﺤﻣ

ﮏﺗ :ﻢﯾراد ﺶﻨﮐاو

𝒗𝒗

_𝟏𝟏

𝑨𝑨

_𝟏𝟏

+ 𝒗𝒗

_𝟐𝟐

𝑨𝑨

_𝟐𝟐

⇔ 𝒗𝒗

_𝟑𝟑

𝑨𝑨

_𝟑𝟑

+ 𝒗𝒗

_𝟒𝟒

𝑨𝑨

_𝟒𝟒

𝒗𝒗

_𝟎𝟎

𝑨𝑨

_𝟎𝟎

+ 𝒗𝒗

_𝟔𝟔

𝑨𝑨

_𝟔𝟔

⇔ 𝒗𝒗

_𝟎𝟎

𝑨𝑨

_𝟎𝟎

+ 𝒗𝒗

_𝟖𝟖

𝑨𝑨

_𝟖𝟖

2 1 4 3 2

1

4 3

2 1

4 3

1( ) _{ν ν} ^{ν ν ν ν}

ν

ν + − −

= _total

P P

X X

X T X

K

6 5 8 7 6

5 7 8

6 5

7 8

2( ) _{ν ν} ^{ν ν ν ν}

ν

ν + − −

= _total

P P

X X

X T X

K

ﻫ شور زا هدﺎﻔﺘﺳا ﺎﺑ ﺪﯾﺎﺑ ﻪﻟدﺎﻌﻣ ود ﻦﯾا يﺎ . دﻮﺷ ﻞﺣ نﺎﻣﺰﻤﻫ ترﻮﺻ ﻪﺑ يدﺪﻋ

𝐥𝐥𝐥𝐥𝑲𝑲_{𝑷𝑷𝟏𝟏} = −∆𝑮𝑮_𝟏𝟏^𝟎𝟎

𝑹𝑹_𝒖𝒖𝑻𝑻 , ∆𝑮𝑮_𝟏𝟏^𝟎𝟎 = −�𝒈𝒈_𝟏𝟏^𝟎𝟎𝒗𝒗_𝟏𝟏 − �𝒈𝒈_𝟏𝟏^𝟎𝟎𝒗𝒗_𝟏𝟏 + �𝒈𝒈_𝟑𝟑^𝟎𝟎𝒗𝒗_𝟑𝟑 + �𝒈𝒈_𝟒𝟒^𝟎𝟎𝒗𝒗_𝟒𝟒 𝐥𝐥𝐥𝐥𝑲𝑲_{𝑷𝑷𝟐𝟐} = −∆𝑮𝑮_𝟐𝟐^𝟎𝟎

𝑹𝑹_𝒖𝒖𝑻𝑻 , ∆𝑮𝑮_𝟐𝟐^𝟎𝟎 = −�𝒈𝒈_𝟎𝟎^𝟎𝟎𝒗𝒗_𝟎𝟎 − �𝒈𝒈_𝟔𝟔^𝟎𝟎𝒗𝒗_𝟔𝟔 + �𝒈𝒈_𝟎𝟎^𝟎𝟎𝒗𝒗_𝟎𝟎 + �𝒈𝒈_𝟖𝟖^𝟎𝟎𝒗𝒗_𝟖𝟖

Advanced Numerical Methods 19

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



11 :

ﺖﻈﻠﻏ ﯽﻟدﺎﻌﺗ

يﺎﻫ ﺶﻨﮐاو ﺮﯾز

ار ياﺮﺑ رﺎﺸﻓ

1 atm و

يﺎﻣد 3000 K

ﻪﺒﺳﺎﺤﻣ ﺪﯿﻨﮐ

. رد

عوﺮﺷ .دراد دﻮﺟو N₂ لﻮﻣ ^0.5 و O₂ لﻮﻣ ^0.5 ، CO₂ لﻮﻣ ¹ ﺶﻨﮐاو

ﺎﺑ هدﺎﻔﺘﺳا زا

ﻪﺟرد ﺶﻨﮐاو

ε1 2 و يﺎﻫ ﺶﻨﮐاو ε قﻮﻓ

رد ﻂﯾاﺮﺷ ﯽﻌﻗاو

ار ناﻮﺗ ﯽﻣ ترﻮﺻ ﻪﺑ

ﺮﯾز :ﺖﺷﻮﻧ

ﻦﯾاﺮﺑﺎﻨﺑ :

CO₂ CO+0.5O₂ 0.5O₂ +0.5N₂  NO

CO₂ +0.5O₂ +0.5N₂ (1-ε₁)CO₂+ ε₁CO+(0.5-ε₂/2 +ε₁/2 )O₂+(0.5-ε₂/2)N₂ ε₂NO

CO₂ (1-ε₁)CO₂+ ε₁CO+ε₁/2 O₂

0.5O₂+ 0.5N₂ (0.5-ε₂/2 )O₂+ (0.5-ε₂/2 )N₂ +ε₂ NO

Advanced Numerical Methods 20

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



11 ) ﻪﻣادا (:

ﺮﺴﮐ ﯽﻟﻮﻣ

يﺎﻫ ﻪﻧﻮﮔ رد

لﺎﺣ لدﺎﻌﺗ

ترﻮﺻ ﻪﺑ ﺮﯾز

ﺖﺳا :

زا لواﺪﺟ طﻮﺑﺮﻣ

ﻪﺑ يﺎﻫ ﺖﺑﺎﺛ لدﺎﻌﺗ

ياﺮﺑ يﺎﻫ ﺶﻨﮐاو

ﻪﯾﺰﺠﺗ ﯽﻟدﺎﻌﺗ

درﻮﻣ ﺮﻈﻧ

ﻢﯾراد :

𝑵𝑵_{𝒔𝒔𝒕𝒕𝒔𝒔} = 𝟐𝟐 + 𝜺𝜺_𝟏𝟏⁄𝟐𝟐 𝑿𝑿_{𝑪𝑪𝑪𝑪𝟐𝟐} = 1−ε₁

𝟐𝟐+ ⁄𝜺𝜺_𝟏𝟏 𝟐𝟐 ; 𝑿𝑿_{𝑪𝑪𝑪𝑪} = ε₁

𝟐𝟐+ ⁄𝜺𝜺_𝟏𝟏 𝟐𝟐 ; 𝑿𝑿_{𝑪𝑪𝟐𝟐}= 1−ε₂ +ε₁

𝟒𝟒+𝜺𝜺_𝟏𝟏

𝑿𝑿_{𝑵𝑵𝟐𝟐} = 1−ε₂

𝟒𝟒+𝜺𝜺_𝟏𝟏 𝑿𝑿_{𝑵𝑵𝑪𝑪} = ε₂

𝟐𝟐+ ⁄𝜺𝜺_𝟏𝟏 𝟐𝟐

𝑻𝑻𝒕𝒕𝒈𝒈𝐊𝐊_{𝐏𝐏𝟏𝟏} 𝟑𝟑𝟎𝟎𝟎𝟎𝟎𝟎 𝐊𝐊 = −𝟎𝟎. 𝟒𝟒𝟖𝟖𝟎𝟎 𝐊𝐊_𝐏𝐏 = 𝟎𝟎. 𝟑𝟑𝟐𝟐𝟎𝟎𝟐𝟐 𝑻𝑻𝒕𝒕𝒈𝒈𝐊𝐊_{𝐏𝐏𝟐𝟐} 𝟑𝟑𝟎𝟎𝟎𝟎𝟎𝟎 𝐊𝐊 = −𝟎𝟎.𝟗𝟗𝟏𝟏𝟑𝟑 𝐊𝐊_𝐏𝐏 = 𝟎𝟎. 𝟏𝟏𝟐𝟐𝟐𝟐𝟐𝟐

Advanced Numerical Methods 21

لدﺎﻌﺗ ﺖﺑﺎﺛ Equilibrium constant )

(

لﺎﺜﻣ



11 ) ﻪﻣادا (:

رد ﻪﺠﯿﺘﻧ 𝐊𝐊_{𝐏𝐏𝟏𝟏} =

𝜺𝜺_𝟏𝟏

𝟐𝟐 + 𝜺𝜺_𝟏𝟏⁄𝟐𝟐 1−𝜺𝜺_𝟐𝟐+𝜺𝜺_𝟏𝟏 𝟒𝟒 + 𝜺𝜺_𝟏𝟏

𝟏𝟏 𝟐𝟐⁄

1−𝜺𝜺_𝟏𝟏 𝟐𝟐 + 𝜺𝜺_𝟏𝟏⁄𝟐𝟐

= 𝟎𝟎.𝟑𝟑𝟐𝟐𝟎𝟎𝟐𝟐

𝐊𝐊_{𝐏𝐏𝟐𝟐} =

𝜺𝜺_𝟐𝟐 𝟐𝟐 + 𝜺𝜺_𝟏𝟏⁄𝟐𝟐 1−𝜺𝜺_𝟐𝟐+𝜺𝜺_𝟏𝟏

𝟒𝟒 + 𝜺𝜺_𝟏𝟏

𝟏𝟏 𝟐𝟐⁄ 1−𝜺𝜺_𝟐𝟐 𝟒𝟒 + 𝜺𝜺_𝟏𝟏

𝟏𝟏 𝟐𝟐⁄ = 𝟎𝟎. 𝟏𝟏𝟐𝟐𝟐𝟐𝟐𝟐

لﻮﻬﺠﻣ ود و ﻪﻟدﺎﻌﻣ ود ﯽﻄﺧﺮﯿﻏ

�𝜺𝜺^𝟏𝟏 = 𝟎𝟎.𝟑𝟑𝟎𝟎𝟑𝟑𝟔𝟔 𝜺𝜺_𝟐𝟐 = 𝟎𝟎.𝟎𝟎𝟎𝟎𝟔𝟔𝟒𝟒

CO₂ +0.5 O₂ +0.5 N₂ 0.6264 CO₂+ 0.3736 CO+0.6587 O₂+0.4717 N₂ + 0.0564 NO