Combustion of solid fuels
• Combustion of solid fuels is constituted by the following steps:
• Drying and heating of the solid fuel particles up to the temperature at which
begins the release of volatiles
• Volatilization of solid fuel particles and ignition; during the release of
volatiles, each solid particle originates a residual char particle
• Oxidation of the volatile matter (homogeneous reactions) and char
(heterogeneous reactions)
• There is no well defined sequence for the previous steps, e.g., the heating of the
particle may occur simultaneously with the volatilization and oxidation of the
volatiles, and oxidation of the char.
• The drying and heating of the particles are endothermic processes controlled by
heat and mass transfer, and depend on the temperature and particle size. The
amount of heat depends on the moisture present in the particles, their size and
properties, e.g., mass diffusivity, specific heat and thermal conductivity.
Solid fuel combustion
Combustion
Combustion of solid fuels
• Drying starts when the temperature of the particles reaches about 105 ºC, i.e.,
when moisture vaporizes and is released.
• The drying and heating of the particles cause physical changes, e.g., the phase
change of the moisture present in the particle may originate cracks sufficient to
break the particle in case the vapour is not released fast enough.
• Volatilization (release of volatile matter) in the coal begins for temperatures in the
range 350ºC – 400ºC. The amount and nature of volatiles depend, e.g., on the
heating rate, final temperature, residence time, particle size. The volatiles consist
mainly of H2O, CO2, CO, H2, light hydrocarbons (mainly CH4) and tars.
• For sufficiently large coal particles, combustion begins with the ignition of the
volatiles. There is experimental evidence that below a critical particle diameter,
heterogeneous combustion of the particles may begin before or in parallel with
the release and combustion of volatiles.
Solid fuel combustion
Combustion
Combustion of solid fuels
• The char is produced after the volatile matter has been driven from the coal
particle and burned, and is constituted mainly by carbon and ashes, with small
amounts of hydrogen, oxygen, nitrogen and sulphur.
• The char is often approximately spherical, particularly for small particles, and
may exhibit cracks and pores, as a result of the release of volatiles, and may be
larger than the original particle.
Char groups
Group I
Group II
Group III
Porosity (%)
Average wall thickness
(mm)
> 70
Variable, 40-70
< 40
<5
>5
>5
Shape
Spherical
Sub-spherical
Angular
2D schematical
representation
Coal char morphological classification
Solid fuel combustion
Combustion
Combustion of solid fuels
• The oxidation of the char is controlled by temperature dependent phenomena,
and three different temperature zones may be identified for coal char:
Zone I – Zone of relatively low temperatures where combustion is chemically
controlled. The reaction rate is low, and most oxidizer molecules diffuse across
the porous structure without reacting. The concentration of oxygen in the pores is
similar to that in the vicinity of the particles.
Zone II – The reaction between the carbon and the oxidizer becomes faster with
the increase of temperature, being controlled not only be chemical kinetics, but
also by diffusion of the oxygen in the porous structure.
Zone III – At high temperature, the reaction at the surface of the particle
becomes very fast, and all the oxygen is consumed there, so that the reaction
rate is controlled by diffusion of oxygen molecule in the vicinity of the particle to
its surface.
Solid fuel combustion
Combustion
Combustion of solid fuels
Solid fuel combustion
Combustion
Heterogeneous reactions
• Heterogeneous reactions involve chemical species in different phyical states,
e.g., gas-solid reactions. The overall process os gas-solid reactions may be
subdivided as follows:
1. Transport of the reactant molecules to the surface by convection and/or
diffusion.
2. Adsorption of the reactant molecule on the surface.
3. Elementary reaction steps involving various combinations of adsorbed
molecules, the surface itself, and gas-phase molecules.
4. Desorption of product molecules from the surface.
5. Transport of the reactant molecules away from the surface by convection
and/or diffusion.
•
If a reactant molecule A is weakly adsorbed, then
R  k T  Α 
•
If the reactant molecule A is strongly adsorbed, then
R  k T 
•
If the reactant molecule A is weakly adsorbed, while the product molecule B is
strongly adsorbed, then R  k T  Α B
Solid fuel combustion
  
Combustion
Combustion of a carbon particle
• Combustion of a carbon particle illustrates the combustion of solid fuels.
• At the surface of the carbon particle the following reactions may occur,
depending mainly on the temperature of the surface:
C + O2  CO2
2C + O2  2CO
C + CO2  2CO
C + H2O  CO + H2
• The main product formed at the surface is CO, which diffuses away from the
surface through the boundary layer where it combines with the inward-diffusing
O2, according to the following global homogeneous reaction:
CO + ½ O2  CO2
• The surface of the char particle is porous, and there is diffusion across the
surface during the oxidation process.
Solid fuel combustion
Combustion
Combustion of a carbon particle
• Simplified theoretical models for the combustion of a carbon particle are based
on the previous global reactions, and generally assume that the surface is
impervious to diffusion.
• These models may be classified as one-film, two-film or continuous-film models,
depending on the simplifying assumptions about chemical kinetics on the surface
and on the gaseous phase.
• In the one-film models there is no flame in the gaseous phase, and the maximum
temperature occurs at the carbon surface.
• In the two-film models the flame front lies at a certain distance from the surface,
where the CO produced at the surface reacts with incoming O2 to form CO2.
• In the continuous film models the flame region is distributed within the boundary
layer, rather than occurring in a sheet.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
• Simplifying assumptions
1. The burning process is quasi-steady
2. The spherical carbon particle burns in a quiescent, infinite ambient medium
that contains only oxygen and an inert gas, such as nitrogen. There are no
interactions with other particles, and the effects of convection are ignored.
3. At the particle surface, the reaction C + O2  CO2 prevails. In general, this
assumption is not particularly good since carbon monoxide is the preferred
product at combustion temperatures, but eliminates the problem of how
and where the CO oxidizes.
4. The gas phase consists only of O2, CO2 and inert gas. The O2 diffuses
inward, reacts with the surface to form CO2, which then diffuses outward.
5. The thermophysical properties of the gaseous phase, lg, cp,g and rDM , are
constant, and the Lewis number is equal to one.
6. The carbon particle is impervious to gas-phase species.
7. The particle is at uniform temperature and radiates as a grey body to the
surroundings, while the medium is transparent to radiation.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
• The main objective of the model is to determine the mass burning rate of the
carbon, m C , and the surface temperature, Ts. Intermediate values of interest are
the mass fractions of O2 and CO2 at the carbon surface.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
• Mass balance at the surface of the particle
 C rs   m
 CO rs   m
 O rs 
m
2
2
• Mass balance at an arbitrary radial position
 C rs   m
 r   m
 CO r   m
 O r 
m
2
2
• The CO2 and O2 flow rates can be related to the stoichiometry associated with
the reaction at the surface:
1 kg C + s kg O2  (s +1) kg CO2
 O2  s m
C
m
 CO2  s  1 m
C
m
• Conservation equation for the mass fraction of oxygen:


1 d
1 d  2 M dyO2
2
r r D
r
r
u
y

r O2
2 dr
2 dr 
dr
r
r

Solid fuel combustion




with yO2(rs)=yO2,s and yO2(∞)=yO2,∞
Combustion
Combustion of a carbon particle
One-film model
• This yields, after some algebra,
y

 1  O 2 ,
s
m C  4 rs rD M ln 
yO 2 , s

 1
s







• To find yO2,s we assume that the heterogeneous reaction C + O2  CO2 is first
order with respect to O2 and the reaction rate RC [kg m-2 s-1] is given by
RC  kC M C O 2 s
with the reaction rate constant given by
 E 
kC  A exp   a 
 RoTs 
The relation between the burning rate and the reaction rate of carbon at the
surface is given by
m C  4 rs2 RC
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
The molar concentration of oxygen may be converted to mass fraction as follows:
O2 s 
pO 2
RoTs
 xO 2 , s
p
M
p
 yO 2 , s mist
RoTs
M O 2 RoTs
Combining the previous equations yields
m C  4 rs2 kC M C
M mist p
yO ,s  K kin yO2 ,s
M O2 RoTs 2
where the kinetics parameter Kkin depends on the pressure, surface temperature
and particle radius.
• Solving the previous equation for yO2,s and substituting into the equation for the
 C yields a non-linear equation that may be solved to find m C .
burning rate m
Alternatively, we may use an electrical circuit analog as follows:
m C 
yO , s  0  yO
2
1
K kin
2
Rkin
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
The expression for m C derived from the conservation equation for the mass fraction
of oxygen may be manipulated to yield
m C  4 rs rD M ln 1  Bo,m 
with
Bo,m 
yO2 ,  yO2 ,s
s  yO2 ,s
Expanding the logarithm in a Taylor series and truncating the series by retaining only
the linear term yields the following approximation
 yO ,  yO ,s  yO ,  yO ,s yO
2 
2
2
m C  4 rs rD M  2
 2

 s  yO ,s   s  yO ,s  Rdiff
2

 
2

 4 r rD M 
s


Note that yO2,s appears in Rdiff, so that the relation between yO2 and m C remains
non-linear.
Since the burning rate derived from chemical kinetics must be the same as that
derived from mass transfer considerations alone, a two-resistor series circuit results.
Taking the potentials to be O2 mass fractions, the carbon flows from a low potential
to a high potential, i.e., in a direction opposite to that of O2.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
The burning rate of the carbon particle may be determined from the electrical
analogue as follows:
m C 
m O 2
s

yO 2 ,   0
Rkin  Rdiff
with
Rkin
M O 2 RoTs
1


K kin 4 rs2 kC M C M m ist p
Rdiff 
s  yO 2 , s
4 rs rD M
An iterative procedure is still needed to determine the solution, because Rdiff
includes the unknown yO2,s.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
• The value of one of the resistances may be dominant, depending primarily on
the particle temperature and size. In fact,
Rkin
s
RoTs rD M 1

Rdiff s  yO 2 , s M mist p kC rs
If Rkin/Rdiff << 1, the combustion rate is controlled by diffusion. Therefore, the
mass fraction of oxygen at the surface of the particle is close to zero. This
happens when kC, rs, p and/or Ts are high (note that even though Ts appears in
the denominator of the equation above, its influence on kc is dominant).
If Rkin / Rdiff >> 1, the combustion rate is controlled by kinetics, so that the
concentration of oxygen at the surface of the particle is very close to that far
away from the particle. This occurs when kC, rs, p orTs are low.
Solid fuel combustion
Combustion
Combustion of a carbon particle
One-film model
• Energy balance at the surface
 O 2 hO 2  m
 CO 2 hCO 2  qs i  qs  f  qrad
m C hC  m
qs-i – heat transfer rate by conduction from
the surface to the fluid (qs-i = 0) under
steady state conditions.
qs-f - convective heat transfer rate from the
surface to the fluid.
The equation above may be expressed as
follows:
m C Q p  lg 4 rs2
dT
dr
r  rs
Solid fuel combustion

4
  s 4 rs2 Ts4  Tsurroundig
s

Combustion
Combustion of a carbon particle
One-film model
• The temperature derivative at the surface may be obtained from the
conservation equation for energy. This equation is identical to that derived for
the droplet evaporation model, which yields
dT
dr
r  rs
Z m C  T  Ts exp  Z m C / rs 
 2 

rs  1  exp  Z m C / rs  
with Z = cp,g / (4 lg)
Inserting into the previous equation yields

  m C c p, g  
 
 exp 
 4 l g rs  


 T  T    4 r 2 T 4  T 4
m C Q p  m C c p , g 
s
s
s
surroundings
 s 



1  exp   mC c p, g  
 4 lg rs  






This equation contains two unknowns: m
 C and Ts. To fully solve the carbon burning
 C is
problem, simultaneous solution of this equation and of the equation for m
 C to yO2,s
needed. If both kinetics and diffusion play a role, the equation relating m
is also required.
Solid fuel combustion
Combustion
Combustion of a carbon particle
Two-film model
• The two-film model describes more realistically the chemical and physical
processes involved in carbon combustion. In particular, the carbon oxidizes to
CO rather CO2.
Solid fuel combustion
Combustion
Combustion of a carbon particle
Two-film model
• The mass flow rates of the various species ca be related by simple mass
balances at the particle surface and at the flame front.
m C  m CO  m CO 2 ,int
(at the surface)
m CO  m O 2  m CO 2 ,int  m CO 2 , ext
(at the flame front)
yielding
m C  m CO 2 , ext  m O 2
We further known that
1 kg C + ss kg CO2  2 (ss +1) kg CO
1 kg C + sf kg O2  (sf +1) kg CO2
with
ss 
M CO 2
MC
sf 
M O2
MC

Solid fuel combustion
M CO 2  M C
MC
 ss  1
Combustion
Combustion of a carbon particle
Two-film model
• The mass flow rates of the various species can be related to the burning rate of
carbon as follows:
m CO 2 ,int  ss m C
m O 2  s f m C  ss  1 m C


m CO 2 , ext  s f  1 m C  ss m C
• The conservation equation for the mass fraction of CO2 in the inner zone is
similar to that for that for the mass fraction of O2 in the one-film model. Hence,
4 r 2 rD M dyCO 2
m C 
ss  yCO 2 dr
with yCO2(rs)=yCO2,s and yCO2(rf)=yCO2,f
Similarly, at the flame surface,
4 r 2 rD M dyCO 2
m C  
ss  yCO 2 dr
Solid fuel combustion
with yCO2(rf)=yCO2,f and yCO2(∞)=0
Combustion
Combustion of a carbon particle
Two-film model
• The conservation equation for the mass fraction of N2 may be expressed as
4 r 2 rD M dyN 2
m C 
yN 2
dr
with yN2(rf)=yN2,f and yN2(∞)= yN2,∞
• Integration of the previous equations gives:
yCO 2 , f

1


rs r f
ss
M
m C  4
rD ln 
yCO 2 , s
r f  rs

1


ss







 yCO 2 , f 
m C  4 rf rD ln 1 

s
s


M
yN 2 , f


m C
 yN 2 , exp 
M 
4

r
r
D

f

Solid fuel combustion
Combustion
Combustion of a carbon particle
Two-film model
• The previous equations along with the following one for overall mass
conservation contain five unknowns: m C , yCO2 ,s , yCO2 , f , yN2 , f and r f
yCO 2 , f  1 y N 2 , f
• The remaining equation is obtained from chemical kinetics. The reaction C +
CO2  2CO is first-order in CO2 concentration, and thus the rate is expressed
in a form identical to that developed for the one-film model:
m C  4 rs2 kC M C
M m ist p
yCO 2 , s
M CO 2 RoTs
or more compactly as
m C  K kin yCO 2 , s
K kin  4 rs2kC M C
Solid fuel combustion
M m ist p
M CO 2 RoTs
Combustion
Combustion of a carbon particle
Two-film model
• The first four equations may be manipulated to eliminate all variables except
 C and yCO2 ,s leading to
m

 C  4 rs rD M ln 1  BCO 2 , m
m

with
BCO 2 , m
 s  1
2 yO 2 ,    s  yCO 2 , s
 ss 

 ss  1
ss  1  
 yCO 2 , s
s
 s 
 C of the last slide) may
• This equation together with the fifth one (equation for m
be solved iteratively to find m
 C (and yCO2,s).
• If the combustion is controlled by diffusion, then yCO 2 , s  0 and the burning rate
m C can be directly evaluated from the equation above.
Solid fuel combustion
Combustion
Combustion of a carbon particle
Particle burning time
• To obtain the surface temperature it is necessary to write and solve energy
balances at the surface and at the flame sheet, as formerly done for the onefilm model.
• For diffusion-controlled burning, the burning time may be determined similarly
to the droplet burning time. The diameter of the carbon particle can be
expressed as a function of time according to the D2 law:
D 2 t   Do2  K B t
with the burning rate constant given by
KB 
8rD M
rC
ln 1  B 
• Setting D=0 gives the particle burning time:
Solid fuel combustion
Do2
tC 
KB
Combustion
Combustion of a carbon particle
Particle burning time
• In the one-film model the transfer number is B ≡ Bo,m, while in the two-film
model is B  BCO ,m with yO ,s  0 or yCO ,s  0 , respectively, for diffusion2
2
2
controlled combustion.
• In order to incorporate in the model the effects of a convective flow over a
burning carbon particle, the film theory of the previous chapter can be applied.
For diffusion-controlled conditions with convection, the mass burning rates are
augmented by the factor Sh/2. For unity Lewis number, Sh=Nu, and thus
m C, diff with convection  Nu

m C, diff 
without convection
2
The Nusselt number may be evaluated with reasonable accuracy using the
correlation given in the previous chapter.
Solid fuel combustion
Combustion