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Lecture 22: Review for Examination 2. https://engineering.purdue.edu/ME525/ME525SP2012Exam2.pdf. https://engineering.purdue.edu/ME525/Homework6Solutions.pdf. https://engineering.purdue.edu/ME525/Homework7Solutions.pdf.

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Lecture 22: Review for Examination 2

https://engineering.purdue.edu/ME525/ME525SP2012Exam2.pdf

https://engineering.purdue.edu/ME525/Homework6Solutions.pdf

https://engineering.purdue.edu/ME525/Homework7Solutions.pdf

https://engineering.purdue.edu/ME525/ME%20525%202013Homework8_Solution.docx

L11: Momentum Equation

The x-component of the momentum equation is:

For steady 1-D flow, neglecting friction

L11: Momentum Equation: Cylindrical Coordinates for Jets

The x-component of the momentum equation is:

A B C D E

A: x momentum flow by axial convection per unit volume

B: x momentum flow by radial convection per unit volume

C: Viscous forces per unit volume

D: Approximate pressure gradient in the x direction

E: Body force in the x direction

• Schwab Zeldovich variable as conserved scalars.
• Consider species equations to define conserved scalars
• Atomic mass is conserved so species equations can
• be multiplied by a fraction representing mass of an
• atomic species and summed over all to provide an
• atomic mass balance equation which will not have
• source or sink terms and will represent a conserved
• scalar.
• Mixture fraction
• Energy as a conserved scalar.

Example Problem: Mixture Fraction Advection-Diffusion

Small velocity, large K

Large velocity, small K

Analysis of Premixed Hydrocarbon Flames

• Species Conservation: Dassumed same for all species.

Burning Rate and Flame Speed Eigen Values

• Substitute and rearrange:

• The laminar flame speed:

Structure of Premixed Ф = 1.0 Methane/Air Flames

Speed ~ 39 cm/s

T, K

O2

CH4

CO2

H2O

CO

H2

OH

H

HO2

H2O2

Structure of Premixed Ф = 0.6 Methane/Air Flames

Speed ~ 13 cm/s

O2

T, K

CH4

H2O

CO2

CO

H2

OH

H

HO2

H2O2

Minimum Ignition Energy for Spark Ignition

• Assume that a spherical volume of premixed gases is heated to Tb by a spark. There is a critical radius Rcrit below which heat losses to the surrounding gas will be too high for the flame to propagate.

Minimum Ignition Energy for Spark Ignition

• Equate energy liberated by reaction (same as energy supplied by spark) to heat lost to surrounding gases to determine Rcrit:

Minimum Ignition Energy for Spark Ignition

• From solution of heat conduction equation for an infinite hollow sphere (see Incropera and Dewitt):

• Using:

Rayleigh Line: Pressure versus Specific Volume

P= 300 kPa

P= 200 kPa

Physically inaccessible region B

P= 100 kPa

Physically

Inaccessible

region A

0.215

0.86 m3/kg

0.43

Detonations and Deflagrations: Comparison

• Typical values for detonations and deflagrations are shown above (Turns, Table 16.1, p. 617). Ma1 is prescribed to be 5.0 for normal shock. For normal shock and deflagration for each P2/P1 a unique normal Ma1 exists based on combined conservation of mass and momentum. For detonation, a range exists based on the heat release rate.

Definition of Detonation Velocity

• The speed at which the unburned mixture enters the detonation wave approximated as one dimensional for an observed riding with the one dimensional detonation wave By definition:and velocity of burned gases = nx,2

Burned

Unburned

nx,1

• nx,2 = c2 =

r2, P2, T2, c2, Ma2

r1, P1, T1,c1, Ma1

• Zeldovich, von Neumann, and Döring in the early 1940\'s independently formulated similar theories of the structure of detonation waves. The structure is shown in the diagram below:

Induction

Zone

Normal

Shock

Reaction Zone

20

P/P1

10

T/T1

r /r1

1

1

1\'

2

1"

• Assume:

1. Laminar, steady, axisymmetric flow, vertical flame axis, axial diffusion is neglected

2. Equal diffusivity, unity Lewis number, conserved scalar approximation

3. Radiation heat transfer treated using radiation heat loss fraction

4. Pressure gradient assumed to be hydrostatic

Conservation Equations: Cylindrical Coordinates, Thin Flame

Conservation of Mass

Conservation of Axial Momentum

Conservation of Species Mass Fractions

Conservation of Energy

Constitutive Relationships: Ideal Gas Law, Lewis Number etc.

Conserved Scalar Equations for Laminar Jet Flame

• Boundary Conditions

At the jet exit plane

Count Unknowns:

Tuesday April 2nd Guest Lecture

Dr. Charles Baukal from John Zink

http://www.purdue.edu/discoverypark/sustainability/events/view.php?id=751

http://books.google.com/books?hl=en&lr=&id=HjurZJYGyuQC&oi=fnd&pg=PR43&dq=Charles+Baukal+John+Zink&ots=wA7kWEEfT_&sig=D3XXtEEW37XSC1IGzulHxmdzu5I