Joseph Martin 01-Biomass Combustion OstwaldDiagram

8/12/2019 Joseph Martin 01-Biomass Combustion OstwaldDiagram

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BIOMASS COMBUSTION 1

COMBUSTION OF BIOMASS FUELS

1. HEATING VALUE,AIR DEMAND AND FLUE GAS PRODUCTION1.1.

Heating Value

Combustion may be defined as a fast spontaneous chemical reaction of the oxidation-

reduction type with large energy release. Such a reaction mainly involves the carbon and

hydrogen elements of any type of fuel as reducers and the atmospheric oxygen as oxidizer.

The main practical objective of a combustion is to transform the chemical latent heat of the

reactants, i.e. the so called Heating Value, into sensible heat. This last can be carried out by

direct radiation to the walls of the combustion chamber and/or by convection using the

reaction products as a heat carrier, or directly converted into mechanical work in a

thermodynamic process.The Heating Value of a fuel is defined as the heat release in a constant pressure process

involving the unit of quantity of the fuel. The Lower Heating Value LHV is defined as the

heat to be removed from the reaction products to obtain a final temperature equal to the initial

temperature of the reactants, assuming that the reaction products remain in gaseous phase, i.e.

that the condensation heat of the water is not available.

This definition corresponds therefore to the difference between the standard enthalpy of the

reactants and the standard enthalpy of the products under normal conditions.

The standard enthalpy of the single chemical species such as C(graphite),H2, O2, ... is zero.

The non-zero standard enthalpy H273of the normal products of a combustion involves CO2

andH2O:

273 2

273 2 vap

H ( CO ) 393500 kJ / kmole

H ( H O ) 241800 kJ / kmole

=

=

The non-zero standard enthalpy of the reactants involves a lot of more or less complicated

chemical species, such as CO, hydrocarbons, alcohols, pure cellulose, ... , which are well

known. Therefore, the standard enthalpy of a fuel completely defined in percentage ofsuch well defined chemical species would be easily computed by linear combination of their

components. The following values are especially useful :

273 a

273

273 4

H ( CH ) 10870 a kJ / kmole ( for char or coke : a 0.2 )

H ( CO ) 111100 kJ / kmole

H ( CH ) 74700 kJ / kmole

=

=

=

By difference between the standard enthalpy of the reactants and the standard enthalpy of the

products, one obtains the following values of severalLHV:


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2

4

0.2

LHV( C ) 393500 kJ / kmole

LHV( H ) 241800 kJ / kmole

LHV( CO ) 282400 kJ / kmole

LHV( CH ) 802400 kJ / kmole

LHV( CH ) 420120 kJ / kmole

=

=

=

=

=

Unfortunately, the actual fuels are often very variable combinations of such species, and their

composition may only be defined as an elementary percentage of single species, without a

thorough knowledge of their actual chemical structure.

A lot of formulae have been proposed to approximate the LHVof several actual fuels, more

especially solid fuels, on basis of their elementary composition. This last can be expressed in

a very convenient form, as a stoechiometry formula written for one atom of carbon :

y x z uCH O N S

For pure and dry biomass fuels of the ligno-cellulosic type, nitrogen and sulphur are usually

negligible and the above formula may be rewritten as follows for a pure (without minerals)

and dry fuel :

y xCH O with y 1.44 and x 0.66

A very good approximation of the Heating Value of such a fuel can be derived from an

equivalent distribution of the actual (unknown) chemical bonds between C,Hand O,based on

pyrolysis data and using simple compounds involving C O= , O H and C H . Thetotality of the oxygen is considered to be distributed on the carbon and on the hydrogen to

form COandH2Obalancing the species as follows :

y x 2 y

x x xCH O CO 0.5 y H O (1 ) CH

1 0.5 y 1 0.5 y 1 0.5 y + +

+ + +.

The remaining hydrogen is then distributed on the carbon to form CH4and a residual coke

CH0.2 as follows :

y 0.2 4

x x 4 y y 0.2

(1 )CH (1 )( CH CH )1 0.5y 1 0.5 y 3.8 3.8

++ + .

One obtains therefore the global equivalence :

y x 2

4 0.2

x xCH O CO 0.5 y H O

1 0.5 y 1 0.5 y

x y 0.2 x 4 y(1 ) CH (1 ) CH

1 0.5 y 3.8 1 0.5 y 3.8

++ +

+ +

+ +

and the following expressions of theLHVof a pure and dry biomass fuel :


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y xCH O

xLHV 400000 100600 y (117600 100600 y ) ( kJ / kmole )

1 0.5 y

x400000 100600 y (117600 100600 y )

1 0.5 y( kJ / kg ) .

12 y 16 x

+ ++

+ ++

+ +

Withy = 1.44andx = 0.66, the valueLHV" is therefore 18500 kJ/kg.

Biomass fuels are often (unfortunately) neither pure neither dry. The mineral matter content

(Mm)may generally be considered as a dilutant of the active species. The moisture content

(Hu) plays not only a role of dilutant, but also a role of active species with a negative LHV,

corresponding to its heat of vaporisation 2500 kJ/kg, consumed during the combustion

process. Assuming that (Mm") and ( Hu" )are expressed as weight ratios to the pure and dry

fuel, the actualLHVof the raw fuel can be easily derived from the valueLHV"of the pure and

dry fuel, as follows :

LHV 2500( Hu )LHV kJ / kg

1 ( Mm ) ( Hu )

=

+ +.

An equivalent formula may be written using the weight ratios (As)and (Hu)to the raw fuel :

LHV [1 ( Mm ) ( Hu )] LHV 2500 ( Hu ) kJ / kg= .

Table 1 illustrates the strong decrease of the Heating Value with the moisture content, more

especially for high mineral content fuels.

0 0.10 0.20 0.30 0.40

0 18500 16400 14300 12200 10100

0.05 17575 15475 13375 11275 9175

0.10 16650 14550 12450 10350 8250

0.15 15725 13625 11525 9425 7325

0.20 14800 12700 10600 8500 6400

Table 1.

Lower Heating Value (kJ/kg) of a biomass fuel versus the moisture and the mineral contents

It is obvious that an efficient biomass combustion system must include a pre-drying device

using any available low potential heat, e.g. heat recovered from the exhaust gases, in order to

maximize the actual availableLHVof the fuel.

(Hu)

(Mm)


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Considering now that only the active matter is an "expensive" consumable, a convenient

"reference" Heating ValueLHVre may be expressed as the heat release corresponding to the

quantity of the pure and dry matter in the raw fuel, i.e. out of the negative LHV of themoisture :

refLHV [1 ( Mm ) ( Hu )] LHV=

The "availability" of this reference Heating Value for any fuel may be expressed as the ratio

between the actual Heating Value LHV of the fuel and its "reference" LHVre . This"availability factor" is a fundamental quality factor of the energy content for a biomass fuel :

LHV

2500( Hu ) 1

[1 ( Mm ) ( Hu )] LHV

With the above value 18500 kJ/kg for LHV", one obtains the following values of this

availability factor as a function of the moisture content and of the mineral content :

0 0.10 0.20 0.30 0.40

0 1 0.985 0.966 0.942 0.910

0.05 1 0.984 0.964 0.938 0.902

0.10 1 0.983 0.961 0.932 0.892

0.15 1 0.982 0.958 0.926 0.880

0.20 1 0.981 0.955 0.919 0.865

Table 2.

LHVrefavailability factor of a biomass fuel versus the moisture and the mineral contents

These values of theLHV availability factorare more significant at an energy viewpoint than

those of the LHV itself, which are combining the energy loss due to the moisture and the

dilution effect of the non fuel content. However, it appears obvious again that an efficient

energy use of a biomass fuel needs an external pre-drying , in order to avoid the in situ

consumption of energy during the combustion itself.

1.2. Air demand and flue gas productionThe here-above symbolic formula of the chemical composition of a ligno-cellulosic fuel is

very convenient to compute the stoechiometric air demand in terms of volumea1

V or in terms

(Hu)

(Mm)


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of massa1

m . The stoechiometric reaction of such a fuel with dry air may be expressed as

follows :

y x 2 2 2 2 2

y 2x y y 2xCH O (1 ) ( O 3.76 N ) CO H O 3.76 (1 ) N

4 2 4

+ + + + + + (1)

where the atmospheric air has been considered to be a dry O2/N2mixture. Taking in account

the normal molecular volume 3N

22.710m / kmole and the involved molecular weights, one

may write :

3

a1 N

a1

y 2x(1 )

4V 108.1 m air / kg fuel 12 y 16 x

y 2x(1 )

4m 137.9 kg air / kg fuel 12 y 16 x

+

=+ +

+

= + +

Similarly, the stoechiometric volume and the stoechiometric mass of the flue gases is easily

computed :

3

f 1 N

y x108.1 130.8 85.4

4 2V m flue gas / kg fuel 12 y 16 x

+ =

+ +

f 1 a1

y 2x1

4m 1 m 1 137.9 kg flue gas / kg fuel 12 y 16 x

+

= + = ++ +

Taking in account the moisture (Hu)and the mineral matter (As), the hereunder expressions

are to be written for the raw fuel :

a1 a1 a1 a1

f 1 f 1 f 1 a1

V [1 ( As ) ( Hu )] V , m [1 ( As ) ( Hu )] m ,

V [1 ( As ) ( Hu )] V 1.262 ( Hu ) , m 1 [1 ( As ) ( Hu )] m .

= =

= + = +

For any actual combustion process, an air-excess factor is needed to ensure complete

combustion. Therefore, the actual air demand and the flue gas production are to be written :

a a1 a a1 f f 1 a1V V , m m , V V ( 1)V= = = + .

The practical values to be derived from the here-above formulae for typical ligno-cellulosic

materials, withy = 1.44andx = 0.66 are :

3 3 3

a1 N N a1 N

3 3 3

f 1 N N f 1 N

V 4.61 m dry air / m dry fuel , m 5.92 kg dry air / m dry fuel

V 5.29 m flue gases / m dry fuel , m 6.92 kg flue gases / m dry fuel

= =

= =

considering the fuel as being dry and without mineral matter.


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2. COMBUSTION STEPS AND COMBUSTION CONTROL2.1. Some fundamentals about biomass fuels combustionThe combustion process for an actual biomass fuel / oxidizer system is of course more

complicated than the behaviour of the "pure gaseous system" or of the "simple

carbon/oxygen" system. For an homogeneous gaseous phase fuel/oxidizer system, such as the

H2/ O2or the CH4/ O2system, the combustion process essentially involves chain reactions

carried by radicals such as OH, H, O, ... . For the heterogeneous solid/gaseous phase C /O2

system, the main combustion process essentially involves CO formation by adsorption-

desorption phenomena on active carbon sites, followed by homogeneous gaseous phase

oxidation. For solid fuels, devolatilisation and thermal decomposition always lead to an

heterogeneous solid-gas fuel system, in which the importance of the gaseous phase depends

on the balance between the so called "volatile matter" content and the "fixed carbon" of thefuel.

The ligno-cellulosic biomass fuels have a relatively high (80 %) volatile content and a

significant (20 %) fixed carbon content. During the combustion, both oxidation processes

(homogeneous and heterogeneous) may be in competition as illustrated fig. 1.

Figure 1

CHyOx

Pyrolysis CHs

CO2 H2O

CnHm CO H2

Oxidation

O2 O2

Oxidation

CHsReduction

CO H2

O2OxidationO2

CO2 H2O

CnHm CO H2

CHs CO H2

CHs

CHs


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This first global approach shows the main interlaced steps of the combustion process.

At a temperature of about 200 C (for the most complex and thus the least stable

compounds) to about 500 C (for the simplest and consequently the most stable

compounds), a thermal cracking or pyrolysis initially decomposes the fuel, forming

radicals. The so formed radicals can lead to gaseous (possibly condensable) recombinationproducts of and/or solids products.

The gaseous products consist of hydrocarbon chains (CnHm,)still being able to include

radicals ; for biomass fuels which contain oxygen, the gaseous products comprise a

fraction of partial oxidation (CO, H2)and of complete oxidation (CO2, H2O)species.

The solid products, only formed by complex fuels, consist of a coke or char, which is

carbon-rich compound (CHs); this coke appears as a porous skeleton image of the

original structure of the fuel ; for pulverized fuels, spongy particles or cenospheres, are

formed, whereas embers result from big pieces of fuels.

The reaction of the pyrolysis products with oxygen then gives place to a first phase of

oxidation. At this level the so called "primary" oxygen is reacting as follows.

The oxidation of the gas compounds involves chain reactions whose active elements

are radicals which are chain carriers responsible of a flame combustion if the

flammability limits are locally met. The flame structure depends on the formation on

the mixture between the fuel gas fraction and the locally available oxygen.

The oxidation of the char needs the adsorption of O2by the active sites of the porous

char surface, forming adsorbed C(O) from which CO is desorbed to burn then in

gaseous phase.

If these oxidation processes are incomplete, one may obtains the following products.

Residual pyrolysis gases or partial oxidation products resulting from the lack of

reactivity or "quenching" at low temperature.

Solid long chain of carbon and hydrogen (in a ratio close to CH0.2) forming soot,

synthesized from the ultimate gaseous residues of the fuel at high temperature without

oxygen.

The carbon of the solid products resulting from oxygen lacking can react with the

surrounding oxidation products CO2andH2O, in a reduction step forming COandH2. If

the temperature is sufficient, the solid phase can thus completely disappear. The so

obtained gas phase includes thus a still combustible fraction .

If necessary, in particular when the combustion process leads to the primary formation of

a still combustible gas, a second oxidation step is needed, using additional or secondary

oxygen, to obtain finally complete oxidation products and therefore to transform into heat

the whole heating value of the fuel.


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From these considerations, it can be said that a successful combustion of a solid fuel such as a

ligno-cellulosic fuel requires at least the following conditions

The gaseous pyrolysis products must form a flammable mixture with air, and this mixture

must be heated (locally or globally) up to the ignition (critical) temperature. Moreover, if

this gaseous mixture is flammable but lacking in air, secondary air must be added withoutfreezing the system, to completely oxidize the gaseous phase emerging from the fuel.

The solid particles of char must form a bed or a suspension able to completely react with

oxygen. This can be achieved efficiently only if the pyrolysis gaseous phase leave the

solid phase free for landing of oxidizing species on the active carbon surface. The locally

formed COmust be burned in a subsequent combustion step, using secondary air.

The above considerations may be illustrated fig. 2, showing the combustion arrangement of a

45 MWthsuperheated steam generator burning wood and bark chips on a moving grate.

Figure 2

The primary air supply is distributed under the grate, to ensure the flame pyrolysis and rich

combustion of the pyrolysis gases (upper part of the grate) and to completely burn the char

(lower part of the grate). the secondary air is distributed at the throat of the combustion

chamber, to completely oxidize the partial oxidation products emerging from the primary

zone. A flue gas recirculation is installed at intermediate zones is for NOx emissions

abatement.

2.2. Checking the combustion by the flue gas analysisAssuming that the combustion of a ligno-cellulosic fuel leads only to gaseous products

excluding residual quenching gases, i.e. assuming that flammability conditions of the


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pyrolysis gases is ensured and that no flying carbon is passing through the system, one may

write the stoechiometry of the fuel combustion (moisture included) as follows :

y x 2 2 2 0 2 1 2 2 1 2 2 2 2CH O z H O w(O 3.76N ) a O a CO a CO b H b H O 3.76wN + + + + + + + + . (2)

The conservation equations of the species may be written :

carbon conservation : 1 2a a 1+ = (3)

hydrogen conservation : 1 2y

b b z2

+ = + (4)

oxygen conservation : 0 1 2 2z

2 a a 2 a b 2 w x2

+ + + = + + (5)

By combining the second and the third of these relations, one eliminates b2andz, what leads

to the remaining system:

1 2a a 1+ = (6)

1 0 1 2

yb 2 a a 2 a 2 w x

2 + = (7)

The stoechiometric coefficients a0, a1, a2, b1 and w are related as follows to the volumic

fractions [ ]' of the dry gas obtained, by the relations :

0 12 0 1 2 1

a a[ O ] , [ CO ] , ... , with a a a b 3.76w

= = = + + + + .

Therefore, the here-above system may be rewritten as follows :

2

2 2 2 2

1[ CO ] [ CO ]

2 y 2x[ H ] [ CO ] 2[ CO ] 2[O ] [ N ]

3.76 2

+ =

+ =

Eliminatingand replacing [N2]'by the closure equation :

2 2 2 2[ N ] 1 [ O ] [ CO ] [CO ] [ H ] = ,

one obtains the following linear equation or compatibility equation between the volume

fractions of the dry flue gases :

2 2 2

y 2x y 2x4.76[O ] ( 2.88 3.76 )[CO ] ( 4.76 3.76 )[ CO ] 0.88[ H ] 1

4 4

+ + + + = .(8)

Considering the gaseous species CO, H2 , CO2 and H2O only existing locally at high

temperature without O2, the following chemical equilibrium must be taken into account :

2 2 2CO H CO H O+ +


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and this equilibrium becomes "frozen" when the temperature is decreasing under ... 850 ... C

at the value :

2

2 2

[ CO ][ H O ]2

[CO ][ H ] .

Considering the coefficients a1and b1as normally small compared to a2and b2, one may take

from (3) and (4) the approximations :

2 2

ya 1 and b z

2 = + ,

what leads to write for the frozen equilibrium :

2 1 2 1

2 2 2 1 1 2

[ CO ][ H O ] a b ay y [CO ]( z ) ( z ) 2

[ CO ][ H ] a b 2 b 2 [ H ]= + = +

from which results the following ratio between2

[ H ] and [CO] :

2[ H ] y 2z

( )[CO ] 4

+=

.

The compatibility equation may thus be considered as a linear relation between the three

independent parameters2 2

[ O ] ,[ CO ] and [ CO ] :

2 2

y 2x y 2 z y 2 x4.76[O ] ( 2.88 3.76( ) 0.88 )[CO] (4.76 3.76 )[CO ] 1

4 4 4

+ + + + + = . (9)

This linear relation describes the plan ( 2 space) of the possible compositions in the of the3

space{[O2]' [CO]' [CO2]'}, as illustrated fig. 3.

Figure 3

[CO]

[O2]

[CO2]

Q

R

P

O


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The summit coordinates of this composition plan are the following :

2 P

2 Q

R

1[ CO ]

y 2x4.76 3.76

4

1[ O ]4.76

1[CO]

y 2x y 2 z2.88 3.76 0.88

4 4

=

+

=

= +

+

The use of the compatibility equation (9) makes it possible to determine any of these three

volume fractions knowing the two others. It is thus possible to obtain a complete diagnosis of

the combustion of a fuel of parameters yandxknown thanks to the measurement of two of

the three fractions.

In the past, it was made use for combustion diagnosis of simple chemical apparatus based on

selective absorbers, or of more expensive instruments based on physical properties to measure

[O2]'and [CO2]or [CO]'and [CO2]' . The electronic miniaturization made it possible today

to develop simple and non expensive apparatus using electrochemical cells based on the

Nernst cell to measure the values of [O2]'and [CO]'. Theses measurement systems includes a

microchip which then makes it possible to display the calculated content [CO2]'in addition

to the directly measured values of the contents [O2]'and [CO]'.

It will be noted finally that if the value of two of the three volume fractions constitutes a

satisfying information for industrial or for checking purposes, it may be desirable, in the case

of reference measurement, to have the redundant measurement of the three volume fractions,

which makes it possible to minimize the uncertainty of the diagnosis by making use of

adequate mathematical methods such as the method of least squares.

2.3. The partial oxidation coefficient and the air excess coefficientThe ratio of the air coefficient w of the actual stoechiometry of the combustion to the air

coefficient

y 2x

(1 )4

+ of the theoretical stoechiometry of the combustion is the air excess

coefficient :

w

y 2x(1 )

4

+

. (10)

Similarly, a partial oxidation coefficient kmay be defined as the ratio :

1

2 1 2

a[CO]k

[ CO ] [ CO ] a a

=

+ + (11)


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Using these definitions and the conservation equations (6) and (7), one may rewrite the

general equation (2) as follows :

y x 2 2 2

2 2

2

2 2

y 2xCH O z H O (1 )( O 3.76 N )

4

y 2 zkCO (1 k )CO k( )H

4

y 2x k y 2z[( 1)( 1 ) (1 )]O

4 2 4

k y 2 z y 2x(1 )( )H O 3.76 (1 ) N

2 2 4

+ + + +

++ +

++ + + +

+ + + +

(12)

form this last equation, one may write the following expression :

2

2

[ O ] y 2 x k y 2 z ( 1)(1 ) (1 )

[CO ] [ CO ] 4 2 4

+= + + +

+

,

which may be rewritten by use of the kcoefficient (11):

2

2

1 y 2 z [ O ] (1 )[ CO ]

2 4 1y 2x

(1 )([ CO ] [CO ] )4

+ +

=

+ +

. (13)

The accurate determination of the air excess coefficient by means of (13) postulates the

measurement of at least two of the three volume fractions {[O2]' [CO]' [CO2]'} by

independent ways, the third of these volume fractions being deduced from both others by the

compatibility equation (9). A better accuracy may of course be obtained by the direct

measurement of the three volume fractions. If one may only measure to two of the volume

fractions, the {[O2]' [CO]'}pair is the most adequate since it provides the most significant

calculation of the third volume fractions by means of the compatibility equation (9).

3. THE OSWALD DIAGRAM AND THE AIR EXCESS OPTIMISATION3.1. The Oswald diagramThe OSWALD diagram of a combustion (fig.2) illustrates in the { [CO2]' [O2]'}coordinates

some of the particular lines related to the flue gases composition. The relations (11) and (13),

can be rewritten in the following forms, linear in {[O2]' [CO]' [CO2]'}:

2

k[ CO ] [ CO ]

1 k =

(14)

2 2

y 2x y 2x 1 y 2z( 1)(1 )[CO ] [( 1)(1 ) (1 )][CO] [O ] 0

4 4 2 4

+ + + + + + = . (15)


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These relations are the equations of the plans k = Cstand = Cst in the {[O2]' [CO]' [CO2]'}3

space. Projecting in the {[CO2]' [O2]'}plan the intersections of the (14) and (15) plans

with the composition plan (9), one obtains straight lines constituting the required remarkable

lines, as illustrated fig. 4 .

Figure 4

The line PQ is the line of complete combustion k = 0 with the following remarkable

points :

point P, characterized by [O2 ]' = 0 and [CO]' = 0, corresponds to a complete

combustion (k = 0)without air in excess (= 1)and is located at the coordinates :

2 P 2 P

1[ CO ] , [ O ] 0

y 2x4.76 3.76( )

4

= =

+

.

point Q, characterized by [CO2]' = 0 and [CO]' = 0, corresponds to an infinite air

excess and thus corresponds to any coefficient k of unburned products and in

particular with that k = 0. Its coordinates are :

2 Q 2 Q

1[ CO ] 0 , [ O ] 4.76 = =

The line PSof stoechiometry = 1, with following remarkable points:

Point P corresponds to k = 0and = 1 , as already described here-above

point S corresponds to k = 1 and = 1 and is located on the [O2]' axis, since its

coordinates are :

2 S 2 S

1 y 2z (1 )

2 4

[ CO ] 0 , [ O ] 3 y 2z y 2x( 1 ) 3.76(1 )2 4 4

++

= =+ + + +

0 5 10 15 20

15

10

5

% CO2

% O2

QS

P

k = 0

k = 1

= 1

LF

M


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The line SQof partial oxidation k = 1, is a portion of the [O2]'axis.

Any normal combustion must lead to a figurative point located in triangle PQS, as close as

possible of point Pif one wishes to minimize the excess of air to obtain flue gases as hot as

possible, while remaining on line PQof complete combustion.

Therefore :

point L, located on line PQ, meets the requirement of complete combustion, but it

corresponds to a important air excess or lean burn combustion. Except for particular

technological reasons (low temperature asked for the use of the flue gases, abatement of

pollutants,), such a point will normally not be retained like standard adjustment.

point M is obtained by decreasing the air excess coefficient compared to that

corresponding to the point Land will be preferred for applications where the character of

not diluted flue gases is desirable, that is to say for thermodynamic reasons(transformation of heat into driving work, transfer of heat,), or for energy reasons

(minimization of flows carrying heat, heat losses at the chimney, minimization of the

ventilation power,).

point Fcorresponds to a rich combustion. Obtained by reducing the air excess near = 1,

it does not meet the requirement of complete combustion and must be rejected. It

corresponds indeed to the production of carbon monoxide CO which is at the same time a

highly toxic compound and a fuel gas from which the LHV is lost.

3.2. The optimization of the air excessThe thermal quality of a combustion decreases at high values of , since the dilution of the

flue gases by the air excess reduces the available temperature. On the one hand, a low

temperature has an unfavorable effect on the kinetics of the combustion itself, and a large air

excess may paradoxally lead to an incomplete combustion characterized by solid or

condensable emissions, by quenching effect. On the other hand, a low temperature of the flue

gases before heat exchange, needs a large size of the heat exchanger and means a low

efficiency of the energy use, by increasing the relative exhaust losses which may be assumed

to be at a constant temperature at the chimney.

The need for a global air excess as small as possible appears therefore as an evidence.

However, using a statistical reasoning about the air distribution, one may conclude that the

unburned fraction kdepends on the mean value and on the local deviations of , i.e. on the

possible misdistribution of the air surrounding the fuel.

This distribution may be characterized by means of a probability densityp( loc)of the locally

defined air excess loc . One of the simplest laws for such a distribution is the rectangular

function:


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loc loc

1p( ) for (1 ) (1 )

2 = < < + (16)

Figure 5

This distribution (fig. 5) has a mean valueand a standard deviation

3.

One may obtain from this distribution function the following results :

if1

1

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Joseph Martin 01-Biomass Combustion OstwaldDiagram