Module 6 Diffusion Flame Lecture 29 Theoretical Analysis Notes | EduRev

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: Module 6 Diffusion Flame Lecture 29 Theoretical Analysis Notes | EduRev

 Page 1


Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_1.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
The Lecture Contains:
Theoretical Analysis
Theoretical Analysis (Contd.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_1.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
The Lecture Contains:
Theoretical Analysis
Theoretical Analysis (Contd.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_2.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis
Consider a 2D diffusion flame,
(Figure 29.1)
Assumptions: 
i. 2D steady laminar inviscid flow.
ii. Velocity above the channel is constant everywhere 
iii. Fuel and oxidizer react in stoichiometric proportion at the flame surface with infinite reaction
rate (Thin flame approximation).
iv. Binary diffusion between participating species.
v. Mass diffusion is along x-direction only.
vi. Unity Lewis number.
vii. Single step irreversible reaction.
viii. Radiation heat transfer is negligibly small.
ix. Constant thermophysical properties.
x. Mass diffusivity of both fuel and oxidizer are the same.
xi. Buoyancy force is neglected.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_1.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
The Lecture Contains:
Theoretical Analysis
Theoretical Analysis (Contd.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_2.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis
Consider a 2D diffusion flame,
(Figure 29.1)
Assumptions: 
i. 2D steady laminar inviscid flow.
ii. Velocity above the channel is constant everywhere 
iii. Fuel and oxidizer react in stoichiometric proportion at the flame surface with infinite reaction
rate (Thin flame approximation).
iv. Binary diffusion between participating species.
v. Mass diffusion is along x-direction only.
vi. Unity Lewis number.
vii. Single step irreversible reaction.
viii. Radiation heat transfer is negligibly small.
ix. Constant thermophysical properties.
x. Mass diffusivity of both fuel and oxidizer are the same.
xi. Buoyancy force is neglected.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_3.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
Conservation equations:
Mass conservation:
Using assumption (ii), we can have,
Axial momentum conservation:
Species conservation equation:
 
Mass fraction of the product can be found from
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_1.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
The Lecture Contains:
Theoretical Analysis
Theoretical Analysis (Contd.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_2.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis
Consider a 2D diffusion flame,
(Figure 29.1)
Assumptions: 
i. 2D steady laminar inviscid flow.
ii. Velocity above the channel is constant everywhere 
iii. Fuel and oxidizer react in stoichiometric proportion at the flame surface with infinite reaction
rate (Thin flame approximation).
iv. Binary diffusion between participating species.
v. Mass diffusion is along x-direction only.
vi. Unity Lewis number.
vii. Single step irreversible reaction.
viii. Radiation heat transfer is negligibly small.
ix. Constant thermophysical properties.
x. Mass diffusivity of both fuel and oxidizer are the same.
xi. Buoyancy force is neglected.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_3.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
Conservation equations:
Mass conservation:
Using assumption (ii), we can have,
Axial momentum conservation:
Species conservation equation:
 
Mass fraction of the product can be found from
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_4.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
By thin flame approximation,
 
Single step irreversible reaction,
Universal concentration variables,
Rate of fuel transport from the centre to the flame surface is equal to stoichiometric rate of oxidizer
transport.
Let  be the mass fraction of the reactant, 
Instead of solving two equations (For fuel and oxidizer), we can solve a single equation as given
below,
This analysis is known as the Burke-Schumann's analysis
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_1.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
The Lecture Contains:
Theoretical Analysis
Theoretical Analysis (Contd.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_2.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis
Consider a 2D diffusion flame,
(Figure 29.1)
Assumptions: 
i. 2D steady laminar inviscid flow.
ii. Velocity above the channel is constant everywhere 
iii. Fuel and oxidizer react in stoichiometric proportion at the flame surface with infinite reaction
rate (Thin flame approximation).
iv. Binary diffusion between participating species.
v. Mass diffusion is along x-direction only.
vi. Unity Lewis number.
vii. Single step irreversible reaction.
viii. Radiation heat transfer is negligibly small.
ix. Constant thermophysical properties.
x. Mass diffusivity of both fuel and oxidizer are the same.
xi. Buoyancy force is neglected.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_3.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
Conservation equations:
Mass conservation:
Using assumption (ii), we can have,
Axial momentum conservation:
Species conservation equation:
 
Mass fraction of the product can be found from
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_4.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
By thin flame approximation,
 
Single step irreversible reaction,
Universal concentration variables,
Rate of fuel transport from the centre to the flame surface is equal to stoichiometric rate of oxidizer
transport.
Let  be the mass fraction of the reactant, 
Instead of solving two equations (For fuel and oxidizer), we can solve a single equation as given
below,
This analysis is known as the Burke-Schumann's analysis
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/Web%20Course/Dr.%20D.P.%20Mishra/Local%20Server/FOC/lecture29/29_5.htm[10/5/2012 4:14:07 PM]
 Module 6: Diffusion Flame
 Lecture 29: Theoretical Analysis
 
Theoretical Analysis (Contd.)
Above equation can be converted into a diffusion equation by substituting
Inner wall exists at  and outer wall at 
The initial and boundary conditions are as follows.
Applying boundary conditions, we obtain a closed form series solution
where,  is the non-dimensional mass fraction of the reactant.
 
The infinite series must have a constant value at the flame surface as given below
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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