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Modeling of Infra-Particle C-CO2 Reaction An Application of the Random Pore Model.

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Dev.Chem. Eng. Mineral Process., 7(5/6), pp.525-536, 1999.
Modeling of Intra-Particle C-CO, Reaction: An
Application of the Random Pore Model
G.S. Liu, A.G. Tate, H.R. Rezaei, A.C. Beath and T.F. Wall
Cooperative Research Centrefor Black Coal Utilization,
Department of Chemical Engineering, The University of Newcastle,
Callaghan, New South Wales 2308, Australia
The Random Pore Model developed by Bhatia et al. was applied to intra-particle
gasijication reactivity of carbon with carbon dioxide in present contribution. Instead
of the conventional n-th order equation, the Langmuir-Hinshelwood equation, which
treats reaction to be an adrorption and desorption two-step reaction, was employed
as an expressionfor the carbon and carbon dioxide surface reaction in the model. A
balance between gas dirffusion and consumptionfor each position in the particle was
established as derived in the Thiele analysis. Numerical calculations were carried out
to solve the integrated equations. The results show that a big difference of reaction
rate is exhibited between using the n-th order rate expression and the LangmuirHinshelwood (L-H) equation. The results also show the distributions of C02 partial
pressure, surface area, reaction rate and carbon conversion within the particle
during the gasification. The overall conversions at diferent structure parameters,
which are correspondent to initialporosity of 0.05, 0.2 and 0.4, are also presented in
the paper. A small deviation of the effectivenessfactors obtained between using the
Langmuir-Hinshelwood and the n-th order equation is indicated with the carbon
burn-off between 045%.
Introduction
The Random Pore Model has been developed by Bhatia and Perlmutter [ 1-31 in order
to account for the relationship between the surface area S and the carbon burn-off X
525
G.S.Liu et al.
during the char gasification. It is assumed that the char gasification reaction with
carbon dioxide is reaction-rate controlled. The critical point of the random pore
model is the consideration of a transformation of pore cylinders fkom non-overlapped
type to overlapped one during the reaction. The relationship between the surface area
S and the carbon conversion X is shown below [ 11:
A Langmuir-Hinshelwood rate expression has been applied to the gasfication
reaction over the last several decades [4-51. The reaction is mechanistically
considered as an adsorption-desorption two-step reaction, and the product gas, CO,
has inhibiting effect on the reaction, which is not shown in the n-th order equation,
.
.
1.e. 1.e. R, = kc .p"co: . Therefore, the intrinsic reaction rate O<g/cmZ.s)is dependent on
both partial pressures of the reactant gas CO, and the product gas COYwhich is
expressed as follows:
In the consideration of the relationship between the partial pressure and the reaction
rate, the Langmuir-Hinshelwood expression doesn't use the uncertain pressure order
n, as indicated in the n-th order equation. Recent studies[4,5] show that the
Langmuir-Hinshelwood expression allows a good understanding of the char
gasification, especially for high pressure conditions.
In the present contribution, the random pore model was applied to the C-CO,
intra-particle gasification, along with the Langmuir-Hinshelwood expression. The
comparison of the distributions of CO, and CO concentration within the particle is
presented. The reaction rate and the surface area with bum-off are also calculated.
The objective of this work is to apply the random pore model, incorparated with the
Langmuir-Hinshelwoodreaction kinetic, to the char gasification with CO, and steam.
526
Modeling of intra-particle C-carbon dioxide reaction
Model
Considering the isothermal chemical reaction of the char particle with carbon dioxide,
the reaction occurs on the surface of the pores contained in the particle. The purpose
of the model is to establish a mass balance of the reactant gas between diffusion and
consumption due to reaction. After the integration of equation (1) and (2), the reaction
rate, defined as carbon conversion rate &dt
(11s)) is composed of the intrinsic
reaction rate ki and the surface area S
where the carbon conversion X,C 0 2partial pressure pcq and the surface area S are a
function of position x and residence time t.
Following the correlation of partial pressure of CO and C 0 2 within the particle
presented by Roberts ef al. [6], let w = 1+ k,(Pco,s
+ P c o ~ , s ~ c /oDco)
,
, and
further let K , = k, I w and K, = (k, - k3flC02/ D,) I w , the reaction rate ki is then
transformed into
which is solely dependent on C 0 2 partial pressure.
To establish a mass balance equation, a modified Thiele modulus is defined as
where S, Vpare the surface area (cm2/g)and the particle volume (cm’/g) respectively.
The effective diffusion coefficient Dco~,which is determined by porous matrix rather
527
G.S.Liu et al.
than gas properties, is dependent on the bulk diffusivity and the Knudsen diffusivity
as well as the pore structure. An alternative for DCo2calculation is given elsewhere
[7]. The COz gas balance equilibrium at each position within the particle is then
formulated as
Boundary conditions for the above differential equation are described as:
dPco, I a!x = dPco ldr = 0 at x = 0 (at centre) and Pco, = Pco2s , Pco = Pco,s at
x = R, (on exposed surface).
Then the analytical solution of the equation (6) is obtained
K,(Pco, - Pco, ,o ) - In
(7)
This gives out the reactant gas C02gradient distribution within the char particle.
Initial condition for equation (7) is: X=O at t=O. Once the local burn off is obtained
from equation (3), the surface area can be calculated using equation (1).
Results and discussion
Numerical simulation has been performed to solve the equation (7) to determine the
COz partial pressure distribution. Runga-Kutta method was used to solve equation (3)
to obtain the local conversion at each residence time. The reaction rate coefficients
are cited from Blackwood et al. [4], which are: kl=5.52x1O9exp(-3l7680/RT)
g/g.atm.s, k2=0.3 l/atm, and k3=14 llatm. Considering pulverized coal char
gasification system, following parameters are used
in the
calculations:
D c o ~ D c ~ 0 . 1 c m 2 / Dp-0.Olcm;
s;
Tp-1500K;So=100m2/g; @=0.5; pp=l .45g/cm3;
Ps,c02‘0.8 atm; and P s , c ~ 0 . 2atm.
528
Modeling of intra-particle C-carbon dioxide reaction
In the following sections the calculation results are presented for the
understanding of the reaction behavior within the particle.
0.90
0.16
-*, 0.80
$ 0.70
g- 0.60
0.14
-P
-C02: n-th
- - - - C02: L-H
g 0.50
u)
=.
0.10
2
;0.08
co: L-H
I
0.40
e
0.12
=
0
;
.-
2
0.30
0.10
0.00
0.02
0.00
0.0
0.2
0.4
0.6
0.8
Radial pcsition. x/Ro
1.0
n-lh
---- Rate: L-H
0.04
- 0.20
d
-Rate:
0.06
K
L
0.0
02
0.4
0.6
08
1.0
Radial position. XlRO
(4
(b)
Figure 1. Predicted distributions of (a) CO2 and CO relative partial pressure and (b)
the calculated reaction rates within the particle using L-Hand n-th order equation,
respectively (Tp=1700K)
1.0 r
7
1
0.014
0.8
0.012
X
.$
.__-----
1 c&.OS.
0.6
-Rate:
--8
t
n-a
---- RaWL-H
0.002
I
8
~49.5
2 Q4.2yb4.5
9
3 c@.4,*=20
0.4
0.2
I
0.0 y
0.0
02
0.6
Radial posilion. nR0
0.4
0.8
1.0
0.0
20
4.0
6.0
8.0
10.0
Tim. s
Figure 2. Distributions of calculated
Figure 3. Overall conversion vs.
reaction rates using L-Hand n-th order
residence time with different particle
equation, respectively (T,=lSOOK)
structure parameters (Tp=I700K)
529
G.S.Liu et al.
Comparison of the two zype reaction rate equations
Using the L-H equation, the distributions of C02 and CO relative partial pressure
(ratio of the partial pressure to the total pressure on the surface) within the particle
were obtained by numerical calculation of equation (7). It is shown in figure l(a) that
the COz partial pressure decreases when the position moves inward from the particle
surface to the particle centre, while the partial pressure of the product gas, CO, partial
pressure increases significantly. The reaction rate exhibits a large variation from the
surface to the centre as seen in figure I@), due to the assumption that the reaction rate
is dependent on both the partial pressure of CO2 and CO. For comparison, the CO2
distribution and reaction rate were also calculated using n-th order equation in the
way similar to the calculation discussed above, which can also be obtained elsewhere
[7]. The intrinsic reaction rate using n-th order rate equation is expressed as:
R,= 5.2exp(-243300 ~r)pO.’
(g/cm2.s). Using this equation, the calculated reaction
CfJI
rate on the particle surface is close to that calculated using L-H equation. As seen in
figure l(a), the C02 partial pressure is similar to the case of using LangmuirHinshelwood equation, while as shown in figure l(b) the reaction rate profile is flatter
due to the rate dependence on C02 partial pressure only.
Reaction rate distributions were also obtained using L-H and n-th order equations
at the particle temperature of 15OOK. No such deviation as indicated in figure l(b)
was observed since the reaction is chemical reaction controlled using either L-H or nth order equation. The reaction rate calculated using n-th order equation however is
higher than that using L-H expression.
Overall conversion
Overall conversion is the total carbon conversion of the particle which is
calculated from the following equation
530
Modeling of intra-particle C-carbondioxide reaction
Overall conversion Xchanging with residence time t is shown in figure 3. From initial
to about 3-5 seconds, the overall conversion increases monotonically with the
increase in the residence time, and then the profile becomes flat with further increase
due to the decrease of the reaction surface area, which will be discussed below. The
overall conversions at different particle structure parameters, N,are also indicated in
figure 3. Assuming the uniform pore size distribution, the particle structure parameter
yois a function of the initial porosity 6 only, shown as follows [3]:
1
yo =-ln(l-Eg)
(9)
As shown in figure 3, the particle with large initial porosity has low carbon
conversion rate. This phenomenon can be explained that with the same initial internal
surface area, the particle with small porosity contains relatively small pores. During
the reaction, the small pores produce more internal surface area than large pores at the
same gasified carbon, which lead to high reaction rate.
Local CO,partial pressure
In the calculation, it is assumed that the diffusion of C02 from the particle surface
to the centre is rapid, in other words, C02 reaction and diffusion are assumed in the
quasi-steady state at any time. As shown in figure 4(a), the C02 partial pressure
distribution is similar to that indicated in figure l(a). As the residence time increases,
the distribution becomes uniform within the particle.
Local su$ace area
Figure 4(b) shows that the local surface area (i.e. the surface area at different
position) changes with the position within the particle at different residence time. The
deviation of the surface area between the particle surface and the centre is observed,
and become significant with the residence time increases. As can be seen in the
figure, at each position the surface area decreases with an increase in residence time
53I
G.S. Liu et al.
due to the collapse of the pore wall. These results are similar to that presented by
Bhatia et a1.[2]in the case of small particle structure parameter fi.
0.81
1
l2
0.80
$ 0.79
5uo 0.78
-----_ ----_
3 0.8 :----
0.77
m
0.76
3
0.75
-l=o.os
---- e 3 . 0 ~
0.74
-
........ e 6 . 0 ~
-..-.. m.os
I
0.73
I
0.72
0.0
0.2
0.4
0.6
Radial position xR0
0.8
0.16
fm
-- -
0.4
*
0.0
L
0.0
1.0
-_-_
-
02
1.0
I
--.
.................. -.__
..-.._
..-..-..
-..
--.---- .- - -.-.___ *-.
.-.
0.6
n
P
I
02
0.4
0.6
0.8
Radial position ~830
,
1.0
I
0.14
-:
0.12
0.10
p!
s 0.08
0
C
2 0.06
a,
= 0.04
0.02
0.00
1I
f
I
0.0
02
0.4
0.6
Radial position xR0
0.8
1.0
::I,,
0.0
0.0
02
0.4
,
0.6
, I
0.8
1.0
Radial position xR0
Figure 4. Disnibutions of (a) C02 relative partial pressure, (b) surjace area, (c)
reaction rate, and (d) carbon conversion within the particle at direrent residence
time. Line notation as in Figure (a) (Tp=1700K)
Local reaction rate
It is very interesting when one looks at the distribution of the local reaction rate
the reaction rates at the
shown in figure 4(c). At the residence time of 0.0 and 3.0~~
particle surface are higher than those at the particle centre, while they are lower at the
532
Modeling of intra-particleC-carbon dioxide reaction
times of 6.0 and 9.0s. This phenomenon can be explained by the distributions of the
surface area and the COZpartial pressure. At times of 0.0 and 3.0s, the distribution of
C 0 2 partial pressure dominates the reaction rates, while at the times of 6.0 and 9.0s,
the distribution of surface area plays an important role in the calculation of reaction
rates. Also it is seen that the reaction rate decreases with the time increases at any
position in the particle.
Local carbon conversion
The carbon conversion at the particle surface is higher than that at the particle
centre at each residence time. At the time of 9.0s, the deviation of the carbon
conversion between the particle surface and the particle centre is about lo%, while at
the time of 3.0s, the difference is 18%. Using n-th order equation, a similar result has
been reported by Bhatia er al. [2].
Effetivenessfactor
Effectiveness factor, which is defined as the ratio of the internal surface area
involved in the reaction to the total surface area, is usually employed in the
determination of the overall reaction rate of the char particle. Once the solution of the
equation (7) is obtained, the effectiveness factor can be calculated using the following
equation:
Where
& is the modified Thiele modulus. For comparison, the effectiveness factors
based on the n-th order equation at the same conditions are also obtained from the
well-known Thiele analysis [7]:
533
G.S.Liu et al.
Table 1 shows the effectiveness factors calculated using both expressions at
different bum-off and the same particle temperature of 1700K. For LangmuirHinshelwood kinetic, the reaction is controlled by combination of reaction and
diffusion control when the carbon bum-offis less than 75%, which corresponds to the
effectiveness factors from 78.4% to 94.2%. It will be transferred to reacton control
with the further increase in the bum-off. At the same conditions, for n-th order
equation, the effectiveness factors are nearly 1 .O over the whole range of the bum-off,
which means the reaction is almost chemical reaction control.
Table I. Comparison of calculated effectiveness factors at direrent burn-of using
Langmuir-Hinshelwood and n-th order equation, respectively
Bum-off(%)
VusingL-H
qusing n-th
I0
I .784
I .989
15
30
40
55
65
75
85
.796
.814
.838
.867
.902
.942
.985
.99
.991
.992
.994
.995
.996
.997
Conclusions
1. A large deviation of reaction rate is shown between using the n-th order equation
and the Langmuir-Hinshelwood expression. A large variation of reaction rate
between the particle surface and the particle centre is also indicated when the
Langmuir-Hinshelwood expression is applied, while the profile of reaction rate
within the particle obtained using the n-th order equation is much flatter. At the
particle temperature of 1500K, reaction rate profiles obtained using both two
methods are uniform, however, the deviation between two approaches still exists.
2. The overall carbon conversion increases monotonically as an increase in the
residence time. This trend becomes insignificant with further increase in residence
time. The conversion rate with high particle structure parameter is higher than that
with low particle structure parameter.
534
Modeling of intra-particle C-carbon dioxide reaction
3. The C02 partial pressure distribution in the particle becomes uniform with an
increase in residence time. The surface area decreases with the time increases,
however, the surface area at the particle surface is lower than that at the particle
centre at any time. The Reaction rate also decreases with the time increases, at the
time of 0.0 and 3.0s, the reaction rate at particle surface is higher than that at the
particle centre, while it is lower at the time of 6.0 and 9.0s. These different trends
are attributed to the distributions of C02 particle pressure and surface area
respectively. The carbon conversion distribution becomes uniform in the particle
as the time increases.
4. A small deviation of the effectiveness factors obtained between using Langmuir-
Hinshelwood and n-th order equation is shown when the carbon burn-off is
between 0-85%. For the Langmuir-Hinshelwood expression, the reaction is
controlled by combination of reaction and &&ion when the bum-off is less than
75%, and then will be transferred into reaction control with the further increase in
bum-off. At the same conditions, for n-th order equation, the effectiveness factor
is nearly 1.O over the whole range of the burn-off.
Acknowledgements
The authors wish to acknowledge the financial support provided by the Cooperative
Research Centre for Black Coal Utilization, which is funded in part by the
Cooperative Research Centres Program of the Commonwealth Government of
Australia.
Nomenclature
Dco
D,-a
kl
k2
k3
Kl
K2
kc
MC
pco
CO diffusion coefficient
C02diffision coefficient
Reaction rate constant
Reaction rate constant
Reaction rate constant
Transformed reaction rate constant
Transformed reaction rate constant
Reaction rate constant
Carbon molecule weight
CO partial pressure
535
G.S. Liu et al.
Pcm
p*.*
R
C02 partial pressure
Total pressure at particle surface
Gas constant
RO
Particle radius
Ri
Intrinsic reaction rate
S
Surface area
SO
Initial surface area
t
Reaction time
T
Particle temperature
VP
Particle volume
X
Distance from particle centre
X
Carbon conversion
Greek symbols
h
Modified Thiele Modulus
Thiele Modulus
4
Initial porosity
Y
O
Particle structure parameter
w
Transformation parameter
Y
Stoichiometriccoefficient
Q
Effectiveness factor
Subscript
0
Initial or particle centre
Local position within the particle
Local
OveraZZ Overall reaction
S
Particle surface
t
Total
es
References
1.
2.
3.
4.
5.
6.
7.
Bhatia, S. K., and Perlmutter, D. D., 1980, A random pore model for fluid-solid reactions: I.
Isothermal, kinetic control, AIChE J., 26,379
Bhatia, S. K., and Perlmutter, D. D., 1981, A random pore model for fluid-solid reactions: 11. Diffusion
and ininsport effects, AIChE J.,27,247
Bhatia, S. K, and Perlmutter, D. D., 1981, The effect of pore structure on fluid-solid reactions:
application to the SO*-lime reaction, AIChEJ., 27,226
Blackwood, J. D., and Ingerne, A. J., 1960, Reaction of carbon with carbon dioxide at high pressure,
Aust. J. Chem., 13, 194
Bliek, A., 1984, Mathematical modeling of a concurrent fixed bed coal gasifier, PhD thesis,
Eindhoven, The Netherlands
Roberts, G. W., and Sattertield, C. N., 1965, Effectiveness factor for porous catalysts. LangmuirHinshelwood kinetic expressions, Ind. Eng. Chem. Fundamentals, 4,288
Satterfield, C. N., 1970, Mass transfer in heterogeneous catalysis, MIT Press
536
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