1. Background

Mixing between fuel and oxygen is one of the most important processes affecting combustion in industrial diffusion flames. The mixing process determines flame shape and flame length, and through them, the flame temperature, radiant heat transfer and ultimately the formation of particulates and pollutants. Even in premixed gas burners, mixing of fresh fuel and comburent with reacted hot products occurring within the flame and between the flame and its surroundings will determine flame properties and stability.

Mixing in flames may be studied by means of mathematical and physical models. Such models have long been a subject of interest to the combustion community, as witnessed by the development of empirical and analytical approaches to describe flame length and shape in the 1960s (see linked CF26). More recently, the emergence of Computational Fluid Dynamics (CFD) as a design tool for combustion engineers provides confirmation of the ongoing importance of mixing in this domain (see linked CF110).

Information on mixing may also be determined from measurements in physical models or directly from measurements in flames themselves. In the latter case, what is actually measured is the local concentration of oxygen, nitrogen, carbon dioxide, carbon monoxide, hydrocarbons, pollutants, solids, ash etc.

It is often difficult to jump directly from gas composition measurements to gain a clear picture of mixing and combustion. It may also be impossible to make direct comparisons between flames on the basis of such data, especially if the fuel or excess air levels are not identical. For this reason, a number of standard mixing and combustion factors have been developed. The numerical value of these factors at any point in a flame may be calculated directly from measured species concentrations of gases and solids.

This Combustion File lists the mixing factors commonly used within the IFRF, and indicates their relative interest to combustion engineers. Linked Combustion Files give the details needed to calculate each of them from in-flame measurements.

2. What are the Principal Mixing and Combustion Factors?

Within the IFRF, two mixing factors and one combustion factor have been defined in order to study mixing and combustion in flames. These factors have been chosen because they are:

–          accurately measurable

–          independent of fuel type

–          applicable to physical  model studies using tracers

–          independent of the nature of the [GLOSS]comburent[/GLOSS] (eg they can cope with oxygen enrichment of the combustion air)

These factors may be calculated from time-mean values of species concentrations measured at a given point.

2.1 Mixing factors

The ‘aerodynamic mixing factor’ (m_a) relates the ratio of fuel and comburent at a given point in the flame to their ratio in the supplies to the burner.

Thus:

m_a = (m_c/m_f)_{flame point} /( m_c/m_f)_burner      (1)

where:

(m_c)_{flame point} = local mass concentration of comburent (kgm^-3) at the flame point

(m_f)_{flame point} = local mass concentration of fuel (kgm^-3) at the flame point

(m_c)_burner = mass flow rate of comburent to the burner (kgs^-1)

(m_f)_burner = mass flow rate of fuel to the burner (kgs^-1)

This factor gives a direct measure of the extent to which mixing has proceeded, and allows direct comparison of mixing in flames with different excess air levels and different fuel characteristics.

m_a takes the value 1 when the fuel and comburent introduced through the burner are completely mixed.

Values of m_a lower than 1 indicate an incomplete mixture at a measuring point having a surplus of fuel (compared to the fully mixed condition). The factor falls to zero when sampling pure fuel {(m_c)_{flame point} = 0}.

Values of m_a higher than 1 indicate an incomplete mixture at a measuring point having a deficiency of fuel (compared to the fully mixed condition). The factor becomes infinite within parts of the furnace having no fuel present {(m_f)_{flame point} = 0}

The ‘stoichiometric mixing factor’ (m_s) compares the ratio of fuel and comburent at a given point in the flame to the [GLOSS]stoichiometric ratio[/GLOSS] for the fuel and comburent concerned.

Thus:

m_s = (m_c/m_f)_{flame point} /( m_c/m_f)_{stoichiometric}    (2)

where:

(m_c/m_f)_{stoichiometric} = mass of comburent required to completely combust unit mass of fuel – the stoichiometric ratio of the fuel and comburent concerned on a mass basis

This factor gives a measure of how far mixing has progressed towards the theoretical (stoichiometric) ratio required for complete combustion of the fuel. It is thus a direct indicator of the shape of a flame, taking account of the fuel characteristics and the level of excess air or oxygen supplied through the burner.

m_s takes the value 1 where a stoichiometric mixture exists. This represents the smallest possible boundary for a flame.

Values less than 1 indicate fuel rich conditions.

Values greater than 1 indicate excess comburent, ie lean conditions

The values of m_a and m_s are independent of the local chemical composition of the fuel and comburent, ie their values are unaffected by the extent to which combustion reactions may have transformed the fuel and comburent into products of combustion. This makes the mixing factors a powerful analytical tool for studying mixing in and around flames. It also means, however, that the mixing factors do not indicate the progress of combustion in a flame. A further factor is available to describe the progress of combustion.

2.2 Combustion factor

Where information is required on the extent to which combustion at a given point in the flame has progressed, the ‘degree of oxidation’ (n) is used.

At a given point in a flame, n is defined as:

n= {(m_c) _reacted}/{( m_c) _reacted + ( m_c) _{needed to complete combustion}}           (3)

where

(m_c) _reacted is the mass concentration of comburent found in reacted species at the point in question (kgm^-3)

( m_c) _{needed to complete combustion} is the amount of comburent needed to complete the reaction of the fuel found at the same point, expressed on the same mass concentration basis (kgm^-3).

When n = 1, combustion is complete. When n < 1, combustion is incomplete. n can never take values greater than 1.

2.3 Combustion lag – unmixedness

If comparisons are made between the degree of oxidation and the stoichiometric mixing factor, with both evaluated on the basis of time mean concentrations, it is often the case that the two are not identical. An obvious example occurs upstream of the flame front, where although fuel and comburent may be mixed, the combustion reaction has not yet begun. Other examples occur within the flame, and may be caused by several factors including turbulent structure and possibly intermittency in the time mean mixing field. For this reason, this difference between the time mean values of the stoichiometric mixing factor m_s and the degree of oxidation n has been labelled unmixedness (u) and defined by [2]:

u = (m_s -n)/ m_s  (for m_s <1)                                (4)

or

 u = 1-n (for m_s >1)                                             (5)

The concept of unmixedness is further developed in related files (CF31, CF134). The definition given in equations (4) and (5) applies at points in flames where the ‘mixed is burnt’ concept is valid (see CF31 for a more detailed explanation).

3. Which factor should I choose?

Each of the factors listed above offers a different perspective on mixing and combustion in a flame.

3.1 Aerodynamic mixing factor m_a

This is the factor to use when the objective is to compare the mixing produced by:

–          different burners or burner configurations,

–          physical models (eg water or cold air models) and their furnace equivalents,

–          mixing in burners operating with different fuels or excess air levels.

In the flue, measured values of m_a different to 1 around a mean value of 1 for the whole flue duct may be indicative of a failure to complete fuel/comburent mixing within the confines of the furnace or boiler.

A consistent bias above or below m_a = 1 throughout the entire flue cross section may indicate:

–          a metering error in the fuel or comburent flow and/or

–          unmetered in-leakage of comburent from sources other than the burner, and/or

–          out-leakage of partially mixed fuel and comburent upstream of the flue.

3.2 Stoichiometric mixing factor m_s

This is the factor to use when wishing to establish or compare:

–          flame length

–          flame shapes

–     [GLOSS]Excess air[/GLOSS]levels at any point within flames or in the flue

Contours of m_s = 1 are useful to visualise the minimum size of a diffusion flame (where mixed = burnt)

In the flue, the value of m_s gives a direct measure of air factor or excess air level.

3.3 Degree of oxidation

This is the factor to use in order to examine the progress of combustion and determine the real flame envelope as opposed to the stoichiometric mixing envelope given in 3.2.

Contours of n = 1 define the flame envelope

Values of n < 1 in the flue are indicative of incomplete combustion. Examination of m_s and m_a will help decide if this is caused by poor mixing or by a shortage of comburent, which may be a sign of air out-leakage from the combustion system prior to mixing.

3.4 Unmixedness

This is a measure of the combustion lag due to the turbulent structure of a high Reynolds number flow field, and to flow intermittency. Where unmixedness is known (eg from measurements in similar flames), the unmixedness factor can be used to estimate the degree of oxidation from the stoichiometric mixing factor using equations (4) or (5). This also provides a means of estimating flame length from time mean mixing measurements in isothermal models.

4. How do I calculate the values of these factors?

All the factors discussed above may be calculated from knowledge of the species concentrations measured at any point in a flame. The relationships needed to undertake these calculations are set out in three linked Combustion Files. One Combustion File has been prepared for each factor (CF116, CF117, CF118). These Combustion Files may be accessed directly from this Combustion File using the links given below.

Sources

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

[2] Fricker N and Leuckel W.  ‘Swirl Stabilisation of high jet momentum natural gas flames, and the optimisation of burner design’. Final report of the NG1 trial series. IFRF Doc nr. F 35/a/4² (1971) (Includes a discussion of unmixedness and gives examples of application of mixing factors)