-
8/16/2019 1979-Fan
1/5
-
8/16/2019 1979-Fan
2/5
334
Table
I.
Variables Relationships
Ind. Eng.
Chem.
Process
Des.
Dev. , Vol. 18, No. 2 1979
emf3 p - p g
minimum fluidization velocity (Davidson and Harrison, 1 971) Urn*=
-
g
5(
1
Em f ) 6idp.11
cm/s
uo
Umf
U b Umf
bubble porosity (Davidson and Harrison, 196 3)
Eb
=
ubble velocity (Davidson and Harrison, 19 63) U b
= U o Um f +
0 .711(gd~ ) '
K = K l K 2 / ( K l K , )
cm/s
1
s
as interchange coefficient (Kunii and Levenspiel, 1969)
Umf
D /
2 g l /
dB d~
5 4
+ 5.85(-
, =
4.5-
€mf
+
D u b ) I 2
K , =
6.78(
d B 3
cmZ/s
B
lateral dispersion coefficient of solids particles (Kunii, 1966)
D =
0 . 1 8 7 ~ b ~ ~ f
€ b
k m f
Table
11.
Numerical Values
of
Fixed Parameters
Used in Computati ons
dB
=
5 , 1 0 , 1 5 c m
R
=
40 cm
=
20, 30, 50 cm
U 85.6-117.0 cm/s
(excess air
=
10-50%)
temperature, 800 C
C =
2.38
X
(mol/cm3)
D
=
1.74 cm2/s
e m f = 0.5
Sh.D
Sh =
2.0
h g=-)
dP
P
=
1.0,
0.75 g/cm3-
d = 0.05 cm
D,,
= 100
D or - cmz/s
Dab
=
D/lOO or
0
cmZ/s
feed rate
F
=
6 g/s
the tre nd of concentration variation is almost independent
of th e bed height. Th e steady-state carbon concentration
is approximately inversely proportional to th e squar e of
th e bed height. On the other hand , the effect of the rate
of excess air is negligible and the bed reaches the s teady
state approximately a t
200 s
when the bubble diam eter is
5 cm. Figure
1
also show s the w ell-known fact (Rengarajan
et al., 1977) th at the s teady -state concentration of carbon
particles is less than 1% by weight.
A
concentration of
1
g/cm 3 roughly corresponds to 1% by weight in th e
present system.
Figure 2 shows the effect of bubble size on the trans ient
carbon concentration. It can be seen that the concen-
tration change is drastically influenced by the bubble size.
Th e average carbon concentration in the bed increases with
an increase in the bubble s ize. This implies tha t a small
bubble operation of the bed is more s table than a large
bubble operation because the bed with a low carbon
concentration can be easily controlled. When th e bubble
diam eter is large, e.g., 15 cm, the bed does not reach a
stab le state because of the insufficient trans fer of gas from
the bubble phase to the emulsion phase. On the other
hand, the small bubble operation reaches a s table s tate
easily.
Th e large bubble operation enhances t he lateral solid
mixing an d, thu s, can minimize th e possibility of gener-
ation of the extremely high concentration near the feeder.
Therefore,
it
is desirable to control the bubble size so t h a t
it is neither too small nor too large.
Figure 3 shows the carbon concentration profiles at
steady s tate with the rate of excess air and th e bubble size
- 3
110
3
2
I
U
e
8
i
0
d : $
0
cm
p
=
I . o g / cm3
r , = 2
0
C r n
L -
20
cm
I
I
30
c m
5
I I 5
em
0.2
2 20 2 0 0
2 zoo00
tim t ~ a c l
Figure 1. Effect of bed height and excess air rate on the transient
average carbon concentration.
I'
=
I S
cm
6 .
4
2 -
5 cm
2
20
200
2
2
2
t i m e t
( I ~ c )
Figure
2.
concentration.
Effect
of
bubble diameter on the transient average
-
8/16/2019 1979-Fan
3/5
Ind. Eng. Chem. Process Des . Dev . ,
Vol.
18, No.
2,
1 9 7 9
335
1 1 I
0
10
2 3
40
r o a i o i
~ s i I i c n
l c m i
Figure 3.
Effect
of
bubble sizeand excess air rate on the steady-state
carbon concentration profiles.
as the parameters . As can be seen from this f igure, the
effect of th e excess air rate is almost negligible in th e range
of the excess air rate between
20%
and
50%.
Figure
3
also
shows that as the lateral mixing of the solids becomes
poorer, an appreciable concentration gradient is generated
along the radius, especially in a small bubble operation.
This phenomenon can sometimes be detr imental, espe-
cially for noniso therm al systems.
As
can be seen, a large
bubble operation drastically reduces the concentration
gradient.
In t he present calculation, we have employed a corre-
lation for th e solids dispersion coefficient derived by Ku nii
(1966) as listed in Table
I.
Note tha t the d ispers ion
coefficient is proportional to the bubble s ize. Hiram a et
al. (1975) experimentally obtained a similar relationship
between th e dispersion coefficient and bubb le size. Solids
mixing in a fluidized bed in both radial and axial directions
is
mainly induced by bubble motion (Toei et al . , 1966),
which is obviously influenced by th e bubble diam eter. In
a fluidized bed where the bubble size is increased by
coalescence along the bed height, the axial dispersion
coefficient should be a function of the bubble size and
change along the bed heigh t. Furthe rmore , the ext ent of
reac tant conversion may be influenced considerably by the
bubble s ize. However, the detail of th e relationship be-
tween th e variable bubble size and the solids mixing is not
well known.
The effect of solids density on the transient average
concentration an d on the s teady s tate concentration are
shown in F igures
4
a n d
5 ,
respectively. In obtaining the
results , we have assum ed th at the solids density does not
change during the reaction. In real systems, however, the
dens ity usually changes slightly with reaction time. As can
be seen in th e f igures, the density difference affects the
concentrations near the center an d wall of the bed. In spite
of these effects , the trend of temporal concentration
variation and the extent of lateral mixing (concentration
profiles along the radius) are l i t t le influenced by the
density.
T he effect of feeding area on th e conc entration profiles
in the s te ady and un steady s tates is shown in Figures 6
an d
7 ,
espectively. T o generate a uniform concentration
profile in a large scale fluidized bed, a m ultipoin ts feeder
is usually used.
As can be seen from Figure 7 , he con-
a x + *
811. 2
Y
L - 3 0 s m
I
- 2 O m
dg
S o c m
1
2
2 2
x1 zoo0
zoo00
t l rn I 1sac1
Figure 4.
Effect of solids density on the trans ient average carbon
Concentration.
Figure
5.
Effect of solids density on the steady-state carbon con-
centration profiles.
6 m
0 2 2 20
200
2ooo
-
n l ( 4
Figure 6.
Effect of feeding area on the transient average carbon
Concentration.
centration gradient can be reduced considerably by en-
larging the feeding area. Th e carbon concentration near
the center of the feeder is approximately proportional to
the feeding area with a n ex ponent of
0.37.
Th e assum ption of isothermal operation becomes less
valid if the solid concentration gradient in radial direction
becomes appreciable. U nder such a condition, the energy
balance, in addition to t he mass balance, must be carried
-
8/16/2019 1979-Fan
4/5
336 Ind. Eng. Chem. P rocess Des . Dev . , Vol. 18, No. 2 1979
I W C D I D
.20%
d g * S O c r n
L -3Ocrn
p . 1 0em3
I . \ /
r a M wlon
r
Ian1
Figure
7.
Effect of feeding area on the steady-state carbon con-
centration profile.
to derive the governing equations. Furthermore, the
distribution in the size of coal particles may have to be
taken into account. Naturally, th e degree of difficulty in
solving the resultant governing equation s
wll
be enhanced.
Our future efforts include considerations of t he effects of
the tem pera ture and particle s ize variation.
Concluding Remarks
The effects of operating variables on the steady-state
and unsteady-state carbon concentration in a shallow
fluidized bed combustor have been investigated by using
the two-phase model of a fluidized bed.
Th e steady-state and unsteady-state concentrations are
influenced profoundly by the bubble s ize. Th e time re-
quired to reach th e s teady s ta te is controlled m ainly by
the bubble size. Th e effect of the other parameters on the
concen trations is negligible when compared with the effect
of bubble size.
Th e change in the particle density with reaction time
has l i t t le effect on the s teady-state and unsteady-state
concentrations. Therefore, it can be assumed tha t th e
density remains constant in th e bed.
Enla rgem ent of th e feeder area is an effective method
in reducing the lateral concentration dis tr ibutions in th e
bed. Th e maximum concentration at the center of the
feeder is proportional to the feeder area with an exponen t
of 0.37.
Appendix
Derivation of the Governing Equations. Consider
a cylindrical shell with a volume of 2arA rL in the shallow
fluidized bed combuster.
For
simplicity, the bubb le phase
and the emulsion phase in this volume element are lumped
separately.
Since the
flow
in the bu bble is assumed to be of the plug
flow, a mass balance of oxygen over an incremental height
AX in this phase is: (accum ulation of oxygen) = (ra te of
oxygen in by convection) (rate of oxygen ou t by con-
vection) (rate of oxygen throug h gas exchange with
emulsion p hase)
or
This is eq
1
in the text.
Since the flow in the axial direction in the emulsion
phase is assumed to be of the com plete mixing type, a mas s
balance of oxygen over this ph ase is: (acc um ulatio n of
oxygen) = (rate
of
oxygen in by convec tion) (ra te of
oxygen out by convection) + (rate of oxygen in by dif-
fusion) rat e of oxygen out by diffusion)
+
(ra te of oxygen
in through gas exchange with bubble phase) (rate of
disappearance by reaction) or
2arArL
(1 b ) emf
2arAr(l tb)Umf(CaO
Cae)+ 2 a r L( 1 b )
NaeIr
2 x ( r + Ar ) L( l
aCae
at
L
' b ) Nae lr+Ar +J 2arAr'?$(Cab Ca J dX
SarArL(1 tb)Ra
Dividing this expression by 2ar Ar L(1 b ) and letting Ar
- gives
where
N,,
is the diffusional flux and is defined as
aCae
N = - D -
ar
e ae
(A-3)
Ra is the reaction r ate
of
oxygen per un it emulsion volume.
Based on the unre acted core model, the reaction r ate for
a single coal particle is
Assuming that the controlling mechanism is gas film
diffusion, we ob tain
r, = adplkgCae (A-5)
Thu s, the reaction rate per unit emulsion volume becomes
R, = ra.n (A-6)
where n is the number of coal particles per unit emulsion
volume. n is related to th e coal concentration, C, by
64-71
C
n = -
where p is the carbon d ensity of coal particles and is as-
sumed co nstant. Substituting eq A-5 and A-7 into eq A-6
gives
(A-8)
k,
R,
=
6 Cae
Substituting eq A-3 and A-8 into eq A-2 gives
emf Cae = rn, (Cao CaJ + (: rDae$)
+
at . L r
Th is is eq 2 in the text.
A mass balance of carbon over the em ulsion phase is:
(accumulation of carbon)
=
(rate of carbon in from fe eder)
+ (rat e of carbon in by diffusion) (rat e of carbon out by
-
8/16/2019 1979-Fan
5/5
Ind. Eng. Chern. Process Des. Dev.,
Vol. 18,
No.
2, 1979
337
D , = effective dispersion coefficient of solids, cm2/s
D,, = effective dispersion co efficient of oxygen in the emulsion
Dab
= effective dispersion coefficient of oxygen in the bubble
phase, cm'/s
diffusion) (rate of disapp earanc e by reaction) or
2.rrrArL 1 q,) 27rrArFr+ 2xrL (1
h ) N,, ,
aC
a t
2 ~ ( r Ar)L(l tb)NcJr+b27rrArL(l tb)Rc
where
at
0 f
F =
z
O
a t r f < r I R
Dividing t his e xpression b y 27rrArL
1
b ) and le t ting A r
- gives
(A-10)
C l a
t
=
h ; rNc) - R ,
where N, is the diffusional f lux and is defined as
aC
ar
N = - D
(A-11)
and w here
b
$F
=
at
0
4
rf
.rrrf2(1 tb)L
O a t r f < r I R
R, is the reaction rat e of carbon an d is given by
(A-12)
k,MC
PdP
R,
= RaMc
= 6 Cae
Sub stitu ting eq A-11 an d A-12 into eq A-10 gives
CCae (A-13)
$ p + ; --rDS--
k,MC
:)
p d
aC
a t
This i s eq 3 in the tex t .
Nomenclature
C
=
carbon concentration in the emulsion phase, g/cm3
C
= oxygen concentration in the emulsion phase, mol/cm3
C a b
= oxygen concentration in the bubble phase, mol cm3
CaO= initial oxygen concentration (feed gas), mol/cm
/
phase, cm'/s
D
=
gas diffusivity in the solid-gas bounda ry, cm2 /s
d B
=-bubble diameter, cm
d
= particle diam eter, cm
f l= solids feeding rate, g/s
g
=
gravitational constant, cm/s2
K
=
gas interchange coefficient,
l / s
L
= bed height, cm
R = radius of the be d, cm
r = radial distance from th e bed cen ter, cm
rf
=
radius of the feeder, cm
Sh
= Sherwood number
t = time, s
U =
superficial velocity of gas, cm/s
v ,f
=
incipient fluidization velocity, cm /s
X = axial distance from th e bed botto m, cm
t b
=
fraction of the bubble phase
p = gas viscosity, g/cm s or
Pa-s
f
= carbon feed rate, g/cm 3 s
= particle density, g/cm3
Literature Cited
Avedesian, M. M., Davidson, J. F.,
Trans.
Inst. Chem.
Eng.
51, 121 (1973).
Davidson, J. F., Harrison, D., Fluidized Particles , Cambridge University Press,
New York, N.Y.. 1963.
Davidson,
J.
F., Harrison, D., Fluidization , Chapter 2, Academic Press, New
York, N.Y., 1971.
Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations ,
Chapter 9 , Prentice-Hall, Englewood Cliffs. N.J.. 1971.
Highley,
J..
Merrick, D., A . I . C h . E . Symp.
Ser. No.
116 67, 219 (1971).
Hirama, T., Ishida, M., Shirai, T.,
Kagaku Kogaku
Rombur Syu. 1, 273 (1975).
Kunii, D., Levenspiel,
O., J .
Chem.
Eng. Jpn. 2,
122 (1969).
Kunii, D., Kagaku Kikai Gijutsu , Maruzen, No. 18, p 161, 1966.
Ucovets, 0.A., The Method of Lines (Review) , English Trans htii n in Difference
Merry, J. M. D., Davidson, J.
F.,
AIChE
J .
51, 361 (1973).
Rengarajan, P., Krishnan,
R.,
Tseng, S. ., Wen,
C.
Y., A.1.Ch.E. 70th Annual
Sincovec,
R .
F., Madse n, N. K.,
ACM Trans. Math. Software
1, 232 (1975).
Toei,
R.,
Matsuno,
R.,
Kagaku Kikai Gijutsu , Maruzen, No. 18. p 135, 1966.
Equations , Vol. I, p 1308, 1965.
Meeting, New York, N.Y., 1977.
Received f o r review May 30, 1978
Accepted December 4, 978
This work was conducted under the sponsorship
of
the Engi-
neering Experiment Station (Energy Study Project) of Kansas
State University.
