1.    Background

The aerodynamic mixing factor compares the mass ratio between comburent and fuel at a point in the flame to the mass ratio of [GLOSS]comburent[/GLOSS] and fuel supplied through the burner. The significance of the aerodynamic mixing factor is discussed in a linked Combustion File (CF115), together with the circumstances where knowledge of this factor might be useful to combustion engineers. The general expression for m_a may also be found in CF115.

The present combustion file presents the relationships needed to calculate the aerodynamic mixing factor (m_a) from data on local gas and solids concentrations as well as information regarding fuel properties and flows of fuel and air to a burner. The concentrations used throughout this Combustion File are assumed to be on a dry basis.

Only hydrogen, sulphur and carbon compounds have been included in these relationships. In some fuels, other materials may also take a significant part in the combustion process. Examples are chlorine, sodium, potassium, calcium, silicon, iron, magnesium and phosphorous which may exist in significant quantities as oxides, sulphates and carbonates as well as HCl. Their inclusion here would render the relationships in this Combustion File even more complex. Users should add additional terms to include for these substances whenever they are measured in significant quantities.

Finally, it should be noted that in pulverised fuel and atomised oil flames, segregation can occur between constituents of the fuel that transform to the vapour phase, and those that remain as solid particles or liquid droplets. The relationships given here ignore such segregation, and are strictly valid only for gas flames.

2. Updates

The relationships and constants given in the original source [1] have been updated using more recent values of atomic masses. Additional hydrocarbon species (up to C₄H₁₀  compared to C₂H₄ in the original) have also been included reflecting the increased use of natural gas, butane and propane since the original equations were derived in the 1960s. The way in which sulphur compounds are treated has been simplified to reflect advances in in-flame concentration measurement techniques.

The opportunity has been taken to use a constant value of 22.4136 Nm³/kmol for the molar volume of all the gases considered in this Combustion File. This leads to some simplification of the constants and equations compared to the originals where a different value of molar volume was given to each component [1]. The resulting changes in the values used to calculate the mixing factors are inferior to 1%.

The constants in the equations have been given to three significant figures, compared to five significant figures in the original work [1]. This seems more than adequate given the recognised uncertainties in flow and flame measurements, as well as the other issues mentioned in section 1 above.

3. Calculation of aerodynamic mixing factor based on oxygen/fuel ratios (m_a(of))

This is the most universal method of calculating the aerodynamic mixing factor, since the data required for this calculation are those most commonly measured in flames. They also have direct equivalents in isothermal model studies.

The aerodynamic mixing factor is expressed in terms of fuel properties and local concentration measurements by:

 

m_a(of) = 4.33(C + H + S)( A^* + D – P) (m`<sub>daf)</sub> )/F(m`_air)                        (1)

 

where the symbols are defined in Nomenclature below,  and where:

 

Free oxygen

A* = 1.43 [O₂]                     (2)

 

Reacted oxygen  

D = 1.43{ [CO₂] + [SO₂] }  + 0.714{ [CO] + [H₂O_c] }        (3)

 

Fuel concentration (unreacted plus reacted based on carbon, hydrogen and sulphur only) 

F = 0.536{ [CO₂] + [CO] } + 1.43 [SO₂] + 0.0899 { [H₂O_c] + [H₂] }

+ 0.716[CH₄] + 1.25[C₂H₄] + 1.34[C₂H₆] + 1.97[C₃H₈] + 2.59[C₄H₁₀]

+ s{ c + S_sol + h }               (4)

 

Water content arising from combustion

[H₂O_c] =

 

5.96(H/C) { [CO₂] + [CO] + [CH₄] + 2[C₂H₄] + 2[C₂H₆] + 3[C₃H₈] + 4[C₄H₁₀] + 1.87(c.s) }

 

– { [H₂] +2[CH₄]  + 2 [C₂H₄] + 3[C₂H₆]  + 4[C₃H₈] + 5[C₄H₁₀] + 11.1(h.s) }           (5)

 

Oxygen originating from the fuel

P = 0.546[CO₂] (O/C)                                 (6)

4. Calculation of aerodynamic mixing factor based on nitrogen/carbon ratios (m_a(nc))

As an alternative to oxygen and fuel, nitrogen and carbon may be used as the basis for calculating m_a. For the purposes of this Combustion File, oxides of nitrogen are assumed to be negligibly small. With these assumptions, the aerodynamic mixing factor is given by:

 

m_a(nc)  = 1.30 CG(m`<sub>daf</sub>)/K(m`_air)                              (7)

where:

Nitrogen concentration

G = 1.25[N₂] – 0.536[CO₂] (N/C)                        (8)

and:

Carbon concentration

K = 0.536{ [CO₂]  +  [CO]  +  [CH₄]  +  2[C₂H₄] + 3[C₃H₈] +4[C₄H₁₀] } +  c.s           (9)

 5. Nomenclature

(Nm³ = m³ at NTP)

Symbol	Meaning	Units
A*	Concentration of free oxygen in the sampled gas	kg oxygen/ Nm3 dry gas
c	Carbon content of solids	kg carbon /kg solids
C	Carbon content of dry and ash free fuel	kg carbon/kg fuel daf
[CO2]	Measured CO2 content in the sampled gas	Nm3 CO2/ Nm3 dry gas
[CO]	Measured CO content in the sampled gas	Nm3 CO/ Nm3 dry gas
[CH4]	Measured CH4 concentration in the sampled gas	Nm3 CH4/ Nm3 dry gas
[C2H4]	Measured C2H4 concentration in the sampled gas	Nm3 C2H4/ Nm3 dry gas
[C2H6]	Measured C2H6 concentration in the sampled gas	Nm3 C2H6/ Nm3 dry gas
[C3H8]	Measured C3H8 concentration in the sampled gas	Nm3 C3H8/ Nm3 dry gas
[C4H10]	Measured C4H10 concentration in the sampled gas	Nm3 C4H10/ Nm3 dry gas
D	Reacted oxygen in the sampled gas	kg oxygen/ Nm3 dry gas
F	Fuel concentration (reacted plus unreacted) in the sampled gas	kg fuel/ Nm3 dry gas
G	Concentration of nitrogen from combustion air in the sampled gas	kg nitrogen/ Nm3 dry gas
h	Hydrogen content of solids in the sampled gas	kg hydrogen/kg solid
H	Hydrogen content of dry and ash free fuel	kg hydrogen/kg fuel daf
[H2]	Hydrogen content of the sampled gas	Nm3 hydrogen/ Nm3 dry gas
[H2Oc]	Water vapour content from combustion	Nm3 water/ Nm3 dry gas
K	Carbon concentration in the sampled gas	kg carbon/ Nm3 dry gas
m`air	Mass flow of comburent (usually air)	kg/s
m`daf	Mass flow of dry ash free fuel	kg/s
ma(nc)	Aerodynamic mixing factor (nitrogen – carbon)	–
ma(of)	Aerodynamic mixing factor (oxygen – fuel)	–
N	Nitrogen content of dry ash free fuel	kg nitrogen/kg fuel daf
[N2]	Nitrogen content of gas	Nm3 nitrogen/ Nm3 dry gas
O	Oxygen content of dry ash free fuel	kg oxygen/kg fuel daf
[O2]	Oxygen content of gas	Nm3 oxygen/ Nm3dry gas
P	Oxygen originating from the fuel	kg oxygen/ Nm3 dry gas
s	Solid concentration in the sampled gas	kg solid/ Nm3 dry gas
S	Sulphur content of dry ash-free fuel	kg sulphur/kg fuel daf
Ssol	Sulphur content of solids in the sampled gas	kg sulphur/kg solids
[SO2]	SO2 content of the sampled gas	Nm3 SO2/ Nm3 gas

Sources

[1] Hemsath, K H, Mixing factors and degree of oxidation: Definitions and formulas for computation. IFRF Doc  G 00/a/1, (1965)