THE MATHEMATICAL MODELING
OF THE FLUIDIZED BED CARBURISING
Daniela Dragomir, Leontin Druga, Liviu Adomnica
Department of Heat Treatments and Surface Engineering,
INTEC, 105 Oltenitei street, Bucharest 4, 75651, Romania
ABSTRACT
This paper describes the mathematical model of the fluidized bed
carburising process and provides the carbon concentration profiles in
fluidized bed carburising,
This mathematical model is based on considerations of fluidized bed
carburising thermodynamic and kinetic taking into account the specificity of
the carbon mass transfer in fluidized bed.
The calculated carbon concentration profiles in low alloyed carburising
steels shown in this paper were obtained for two fluidized bed carburising
atmospheres.
INTRODUCTION
Is well known that the different carburising methods are used for surface
hardening steels parts in view of improving their fatigue strength and wear
resistance.
Among these, the fluidised bed carburising is a special method which
assures high heat and mass transfer.
The masstransfer processes at the gaseous atmosphere - metallic material
interface are accelerated by the destroying of the limit gaseous layer (Nernst).
This takes place by means of the continuous collision and bubbling of the
layer particles with the material surface.
The specific moving and rapid circulation rate of the solid particles under
the influence of fluidising gas and the large solid -gas interfaces provide this
phenomenon.
In this manner at the material surface permanently is assured a chemical
active gaseous flux, characterized by a high chemical potential. Also the
diffusion at interface is improved due to the cleaning and activating of the
surface.
As a result of these, the fluidized bed accelerates both the thermodynamic of
the process because change the gaseous rates of the reactions in gaseous
atmosphere (the particles act like catalyst) and the kinetic of the process,
increasing the active elements concentration at the interface and changing the
adsorption and diffusion.
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In this manner the fluidised bed carburising provides many advantages as
short carburising time, easily producing or switching of the atmosphere,
temperature uniformity in the workspace, technological consumption
reducing, others.
The fluidized bed carburising is a flexible process performed in different
atmospheres (tabel 1), at temperatures in the range 900° .... 1000°C, providing
high carburising rates (0,4-0,6 mm/h).
Table 1 – Carburising atmospheres in fluidized bed [1, 2, 3, 4]
Atmosphere
Atmosphere composition
CH4,C3H8 + CH3OH/N2
CH3OH + CH3COCH3
CH4,C3H8 + N2
CH4,C3H8 + air
CH4 + endo
47%CH3OH + 47%N2 + 6%CH4
85%(50%CH3OH+50%N2) +
15%C3H8
75%(50%CH3OH+50%N2) + 25%CH4
CH3OH+CH3COCH3
60%CH4 + 40%N2
30%C3H8 + 70%N2
20%C3H8 + 80%N2
70%CH4 + 30%air
66,7%CH4 + 33,3%air
25%C3H8 + 75%air
33,3%C3H8 + 66,7%air
CH4 + endo
The heat and thermochemical fluidized bed equipment with different design
can be heated taking into account both technological requests and
economics, as follows: with metallic or nonmetallic resistors, electrodes,
burners, radiant tubes inside or outside the retort, etc.
In the modern fluidised bed furnaces design are provided the exhaust gas
recycling/re- using systems, temperature controllers, oxygen probes and can
be assisted by computer.
THE MASS TRANSFER COEFFICIENT ß IN FLUIDISED BED
CARBURISING
The mass transfer rate in fluidised bed processes is directly determined by
particles and fluidising gas circulation in the bed.
In technical literature [1, 7] are noticed 2 solid particles circulation ways
under the influence of fluidising gas - a chaotic circulation of the individual
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solid particles and a circulation on the whole, which determine three phases:
continuous (emulsion), bubbles and particles clouds.
Taking into account of these phases and temperature, the mass transfer in
fluidised bed takes place in two ways:
1. mass transfer between fluid and particles
- mass transfer of the gas between particles and bubbles (mass transfer of the
gas from the bubbles to the chemical inactives particles, descendingly in
bed);
- mass transfer of the gas through the emulsion phase to the chemical actives
particles (mass transfer through conductive mechanism from the inerts
particles groupes which transport fresh gases to the chemical actives high
particle (clouds of particles) which ascend in the bed);
2. mass transfer between bubbles and interstitial gas (because of bubble
surface permeability is available permanently a substance flux, from inside
to outside of bubble under the influence of pressure gradient existed in
fluidised bed continuous phase).
The carbon mass transfer coefficient in fluidized bed carburising can be
determined as follows:
- using criterial calculations:
1
2D 4D 2
 [
 ( ) ] mf
d
tm
(1)
1
(2)
 bd

1
 bn

1
 nd
2
Vmf
1
D .g
bn  4,5( )  5,85( A 5 ) 4
Db
Db
 nd  6,78(
DA mf Vb
3
)
1
3
(3)
(4)
Db
where:
-  is the total mass transfer coefficient between fluid and particles;
- bn is the total mass transfer coefficient between bubbles and
interstitial gas;
- nd is the partial mass transfer coefficient between clouds phase and
dense phase per unit of bubble volume;
- bn is the partial mass transfer coeffiecint between bubbles phase and
dense phase per unit of bubble phase;
- Db is the bubble diamter;
- Vmf is the maximum fluidizing rate;
- Vb is the bubble rate;
- DA is the axial difussion coefficient;
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- mf is the holes fraction at minimum fluidizing rate.
- in the same way used at conventional carburising, depending on
carburising rate of the foils with 50 m thickness which are carburated at a
given temperature and after different time are removed from the carburiisng
atmosphere and are quenched.
The adsorbed carbon content into foil can be determined gravimetrically or
by means or other analyses (ex. foil electric resistance measuring [1, 6]).
Expressing the carbon flux through the carbon content which get into the
steel in the surface unitt in unit of time, the mass transfer coefficient ß, is:
M
M
[cm/s]
(5)
j
  ( Cp  Cs)   
t
Ft ( Cp  Cs)
where:
- Cp-Cs is the difference between gaseous atmosphere carbon activity, aCgas,
characterised by Cp (carbon potential) and surface steel carbon activity
which is carburised aCotel, characterized by Cs, (carbon content into surface).
- M - the carbon content which is adsorbed in foil in t time, in grams;
- F - is the analysed surface for carbon transfer, in cm2;
- t is analysing time, in sec.
Taking into account of these, the carbon mass transfer coefficient ß
correspond of that carbon quantity which diffuses into the steel surface in
unit of time and surface and influences the surface steel carbon content, Cs,
which is obtained in the carburising process.
The higher the carbon mass transfer coefficient  which is characteristic for
each carburising atmosphere will be, the faster the surface content will
increase and will attain the carbon potential value.
Can be noticed that the mass transfer coefficient obtained in fluidised bed
carburising is twice the mass transfer coefficient obtained in conventional
carburising in similar atmosphere (=1,25x10-5 cm/s for conventional
carburising in endo+CH4, respectively =2,8x10-5 cm/s in fluidised bed),
and the fluidised bed carburising time is much reduced comparatively with
conventional carburising.
THE MATHEMATICAL MODELING OF FLUIDIZED BED
The mathematical modeling of the fluidized bed carburising was performed
by solving the Fick’s second difussion law by means of finite differences
method.
Were assumed that diffusion coefficient is dependent of concentration and
temperature and the carbon transfer coefficient  is calculated for fluidized
bed.
C
 2C
D 2
t
x
(6)
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The initial conditions is:
C(x,t) = C0 at t=0
The boundary conditions are:
C ( x, t )
t 0  0
t
(7)
(8)
C(x,t) = Ce at x =0
(9)
C
(10)
x
The Fick’s second diffusion law for one-dimensional space is solved by
means of building of an two-dimensional network – x and t with steps x
and t. For each node of order i, j of this network coresponds a value of
carbon concentration Ci,j.
The equation (6) allows the following discretizes :
 [Ce  C (0, t )]  D
C
1
 [Ci , j 1  Ci , j ]
x x
(11)
C
1
 [Ci 1, j  Ci , j ]
x x
(12)
 2C
1
 2 [Ci 1, j  2Ci , j  Ci 1, j ]
2
x
 x
The third order boundary condition is now:
Ci , j 
1
D
[ Ce  Ci 1, j ]
D
x

x
(13)
(14)
Was assumed that D has the following relation:
D  ( a  b * c )e

Q
RT
(15)
where: a, b are constants and Q is the activation energy.
Using this in the development of the equation (10), finally were obtained:
C
 2C
C
 D 2  b[ ]2
t
x
x
(16)
with the numerical solution:
t
(17)
Ci , j 1  Ci , j  [b(Ci 1, j  Ci , j ) 2  (a  bCi , j )  (Ci 1, j  Ci 1, j  2Ci , j )]
x
This solution is convergent for:
0
t 1

x 2
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(18)
In the equations (6) …(18) where noted:
- C is the concentration function;
- T is the time;
- D is the diffusion coefficient;
-  is the carbon transfer coefficient;
- Ce is the carbon potential of the atmosphere;
- C0 is the initial concentration of the steel.
RESULTS OF THE MATHEMATICAL MODEL
On basis on the above-mentioned model a flexible program in Turbo Pascal
was developed alowing to simulate the fluidized bed carburising process.
The conditions for simulation were:
- carburising time: 2… 4 h;
- carburising temperature: 925 … 940°C;
- carbon potential: 0,8 .. 1,2%C;
- steel grade: 1.7264 (19MoCr11); 1.7131(18MnCr11)
- carburising atmosphere: endo+CH4, CH3OH/N2
In fig. 1…2 are shown the carbon profiles obtained on 16MnCr5 steel in the
carburising atmosphere above mentioned at T=925°C and t = 3 h.
137
Fig. 1 Carbon profile obtained on 16MnCr5 after carburising at
T=920°C/3h/CH4+endo
Fig. 2 Carbon profile obtained on 16MnCr5 after carburising at
T=920°C/3h/CH3OH/N2
CONCLUSIONS
-the mathematical model described above gives a good description of the
fluidized bed carburising process under certain practical conditions.
-the fluidised bed carburising is a possible alternative for conventional
carburising providing short process times;
- the carbon transfer coefficient  is much higher in fluidized bed carburising
than in the conventional carburising
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incalzire in metalurgie, , Bucuresti , Editura Tehnica, 1986
2. P. Sommer - Carburizing in Fluidized Beds, Heat Treatments of Metals
1987, 1, 7 - 10.
3. Mc. Kenzie R.T. - "Selection of Gaseous Atmospheres in Fluidized Bed
Furnaces for Controlled Heat Treatments, Metalurgia, August 1987, 353 355.
138
4. Fennell G. Anthony - Fluidized Bed Heat Treatments Utilizing Controlled
Atmospheres, Industrial Heating, November 1985, 24 - 25.
5. Edenhofer B. - "Technology, Advantages and Applications of Direct - feed
Atmospheres for Carburising, Heat Treatments of Metals, 1995 3, 55 – 60
6.Grabke H., J., - Kinetik des Gasaufkohlens, HTM 1990 45 (2), 110-118
7. Ivanus Gh. et.al - Ingineria fluidizarii, Bucuresti, Editura Tehnica, 1996
139