A. MEHRA
462
ISOLID (B)
Fig. 1. Schematic representation of mass transfer with instantaneous reaction near the gas-liquid interface (penetration element) in a slurry of “fine” particles.
dissolution itself could be enhanced in a manner similar to that for gas absorption accompanied by an
instantaneous reaction. Sada and co-workers have
subsequently reported a large number of studies; these
include experimental investigations, mostly on the
absorption of sulfur dioxide into suspensions of magnesium hydroxide (Sada et al., 1977b, 1979,1980), and
theoretical extensions for the cases when the bulk
aqueous phase is nor saturated with the dissolved
solid material [finite slurries; Sada et al. (1979)] as
well as when the reaction between the dissolved gas
and solid is fast but not instantaneous
[numerical
solutions; Sada et al. (1977a)]. Some experimental
data have also been reported by Uchida et al. (1978)
on the absorption of lean sulfur dioxide into fine
limestone slurries. All these studies have used the
film model of mass transfer with chemical reaction
to analyse and interpret the experimental data. A
penetration theory analog of the model given by
Ramachandran and Sharma (1969) has been proposed
by Uchida et al. (1981) where they showed that for
equal diffusivities of A and B, an analytical solution
could be found for the problem of a moving reaction
front in the presence of fine, sparingly soluble, solid
reactant particles (using the concept of negative concentrations). The prediction of rate values from this
model have been demonstrated to be within a few
percent of the film model computations. Yagi and
Hikita (1987) have critiqued the work of Sada and
co-workers as well as that of Uchida and co-workers
by indicating that the claimed agreement between
theoretical estimates of rates and the experimental
values in some of their studies (Sada et al., 1977a;
Uchida et al., 1978) is based on the use of unreasonable values of system parameters. Furthermore, Yagi
and Hikita (1987) have proposed that an additional
parameter representing the average interparticle spacing is necessary to characterize the system and that
the notion of a mass transfer coefficient for the particle dissolution process is inappropriate. Sada et al.
(1981, 1983) have reported some more work on the
absorption of sulfur dioxide into aqueous slurries
of magnesium hydroxide and limestone so that there
are two reaction planes produced in the “film”
zone - one due to the reaction between SO2 and
OH-/SO:and the other for the reaction of HSO;
with CO:-. In these studies they proposed that there
are “no particles suspending” in the zone enclosed by
the first reaction plane and the gas-liquid interface
since this zone thickness is of the order of the particle
size itself. In another study (Sada et al., 1984), this
concept of a thin, particle-free zone was extended to
the interpretation of rate data for the absorption of
carbon dioxide-sulfur dioxide into calcium hydroxide
slurries, since, otherwise, the theory was found to yield
rate values larger than the experimentally observed
values.
It is indeed surprising that despite the large number
of studies in the area of gas absorption into reactive
slurries, there are none which focus on or account for
the change in the size(s) of the dissolving particles and
the concomitant efict of this phenomenon on the specific rate itself. This point has been noted by Beenackers
and van Swaaij (1993) in a recent review. These
authors have cited their previous publication
(Beenackers and van Swaaij, 1986) in which they have
indicated that the situation of disappearing particles
may actually occur in practical systems. The only
other paper which makes a critical reference to the
assumption of a fixed, unchanging particle size is that
of Yagi and Hikita (1987) where the authors have
provided an expression for the upper bound for the
enhancement factor in the specific rate by assuming
that all the particles are present in “completely dissolved form” so that the extra reactant contained in
the solid phase can be treated as being “immediately”
available to the liquid phase. It will be shown later
in the current work that even this upper limit is too
high.
It is also clear from the above survey of the literature that most of the studies deal with fast, in fact,
instantaneous, reactions between the reactant derived
from the solid phase and the dissolved solute gas.
Since the reaction is fast, usually, only negligible
amounts of the dissolved gas are ever able to reach the
bulk slurry phase, implying thereby that all the reaction
and particle dissolution occurs near the gas-liquid interface, exclusively. The above observations therefore
imply that changes in particle size(s) and the consequence of thisfor the specific absorption rate have to be
accounted for in the zone near the gas-liquid interface,
i.e. the surface penetration element. As will be shown
later, particles which are fine enough can dissolve to
such an extent that these become significantly smaller
(and may even dissolve completely!) in relation to
their original size, during the time of contact of the
surface element with the gas phase (at the gas-liquid
Gas absorption in reactive slurries
interface). This dissolution may have drastic effects on
the specific rate of gas absorption.
A related problem is the estimation of the particle
size distribution as it evolves in the bulk slurry phase
for a given type of slurry absorber. Consider, for
instance, a batch slurry reactor. The evolution of the
particle size distribution in the bulk slurry, on account
of the particle dissolution, with batch time is important not only for determining the specific rate-batch
time trajectory but also for calculating the overall
extent of conversion of the solid reactant initially
loaded into the slurry reactor with batch time. Tracking this bulk particle size distribution involves particle
accounting within the surface penetration elements
since it is in this zone that dissolution may occur
exclusively; the resulting distribution in the bulk
phase then arises out of the convective mixing of
these elements arriving from and departing to the
gas-liquid interface.
This work was therefore carried out with two objectives in mind, namely (i) to assess the effect of particle
dissolution and the consequent change in particle
size(s) near the gas-liquid interface on the point specific rate of absorption (“point”, here, signifies the rate
at any given point on the rate-batch time curve), and
(ii) to examine the nature of the evolution of the
bulk-phase particle size distribution arising out of
the phenomenon mentioned above in (i) and its effect
on the specific rate as well as solid conversion
trajectories with batch time for a typical batch slurry
contactor. Some of the experimental data and observations from the prior literature have also been
reviewed in the light of the conclusions that emerge
from this study.
Material
elements
463
and population
balances
in penetration
Therefore, if the number concentration (number per
unit volume of liquid phase) of solid reactant particles
of size (radius) between R and R + dR, located at
a distance x from the gas-liquid interface within a surface penetration element, at time t, is given by
n(R, x, t) dR, the species balance for the diffusing, dissolved gaseous solute, A, in the surface element
(liquid) becomes
D
a2cA= f$
Ax-
+k2CACB
while a similar balance for the liquid-phase reactant,
B, which is derived from the dissolution of the solid
particles, yields
a2cB acB
= at +
a.9
DB-
zk2CACB
s
L.
-
‘hDBShR(~B - c,)
Rn(R, x, t) dR.
0
= k,, 4n R’(Cs, - C,)
surface penetration element as well as the bulk slurry.
The use of population balances within diffusion
“films” is relatively novel and has been generally
avoided on account of being highly computation intensive. Wachi and Jones (1991a, b) have used this
approach for modelling precipitation
near the
gas-liquid interface and more recently Saraph and
Mehra (1994) have reported the use of simplified
population balances in their study on “near-the-interface” effects in microphase catalysis.
(2)
The first and second terms in the above equations are
the conventional diffusion and accumulation terms,
respectively, while the third term in both the equations represents a second-order, liquid-phase reaction
where z is the stoichiometric factor for the reaction
A + zB --t products. The last term in eq. (2) accounts
for the amount of B that comes into the liquid phase
by dissolution of the particles and may be derived as
shown below.
The rate at which a single particle of size R supplies
B to the liquid phase is given by
THEORY
In order to account for the changes in particle
size(s) of the slurry reactant with time, due to dissolution near the gas-liquid interface, it is necessary to
use an unsteady-state theory of mass transfer; the
film model cannot be used since it is a steadystate description which allows only for the spatial
variations in the concentration
profiles of the
relevant species. In this study, Higbie’s penetration
theory has been used details of which may be found
elsewhere (Danckwerts,
1970; Doraiswamy
and
Sharma, 1984).
Since, in general, the particles that constitute the
slurry can have a size distribution it becomes imperative to use the population balance approach within the
(1)
(3)
where k,[ is the solid-liquid mass transfer coefficient.
Therefore, the total volumetric rate at which B is
transferred to the liquid phase from all the particles is
given by
R,..
RLU=
I
k,147cR2(Cb- C,) n(R, x, t) dR.
(4)
0
Now, if the solid-liquid mass transfer coefficient for
a single particle, in the above equation, is set to
k,[ = Sh,Ds/R
(5)
the last term in eq. (2) results.
In writing the above equations, it has been assumed
that the particles are spherical and that there are no
“blockage” effects on the diffusion process due to the
presence of solid particles. This is in consonance with
the pseudo-homogeneous
models, which have been
postulated in the previous literature, for such “microheterogeneous” media (Bruining et al., 1986; Mehra,
1988). The overall change in the volume of the total
slurry (i.e. solid + liquid) due to shrinkage/ dissolution of the particles has been neglected as this change
is small for low to moderately loaded slurries. In any
464
A. MEHRA
case, this change in total volume cannot be accounted
for by a pseudo-homogeneous
approach and truly
heterogeneous models for concentrated slurries need
to be formulated for these situations as, for example,
has been attempted by Yagi and Hikita (1987) or in
the context of suspended catalysts, by Karve and
Juvekar (1990). The particles have been taken to be
stationary (non-diffusing) and hence a reasonable
value of the radius-based Sherwood number, ShR, is
1.0, i.e. for an immobile particle suspended in a stagnant fluid (this in fact represents the lowest possible
rate of transfer). This also implies that the concentration gradients of adjacent particles do not overlap
which is consistent with a psuedo-homogeneous
approach that is rigorously valid at “low” hold-ups of
the microdispersed phase; a more detailed discussion
on the implications of pseudo-homogeneity
may be
found in Mehra (19881990). The Brownian diffusivity
of particles of radius, say, 0.1 pm, as estimated from
the Stokes-Einstein equation is about 1 x lo-r2 m”/s
which is much less than the typical liquid-phase molecular diffusion coefficients which are of the order of
1 x lo-’ m2/s. Hence, the assumption of stationary
particles within a penetration element is entirely justified and for this case the Sherwood number will have
no dependence on the particle size R. Also, for typical
particle sizes of l-10 pm, used to make fine slurries,
eq. (5) (with ShR = 1) gives values of k,, which are of
the order of 1 x 10-j to 1 x 10e4 m/s, which are in
agreement with the typical values for this coefficient
indicated by Doraiswamy and Sharma (1984). Incidentally, the values of ksl used by Ramachandran and
Sharma (1969) in their numerical example are too low
while the values used by Doraiswamy and Sharma
(1984) in their solved example seem to be more realistic. Yagi and Hikita (1987) have criticized the use of
a mass transfer coefficient for the solid dissolution
effects and have suggested that, instead, a parameter
representing the average spacing between particles be
used. However, their critique is more applicable to
concentrated slurries and in any case the model that
these authors present predicts enhancement factor
values which are as high as infinity! The modification
of Uchida et al. (1975) for the enhancement of the
dissolution process has been neglected in the current
work in formulating eqs (1) and (2) as this effect is
likely to have a significant impact only ‘when q = Cy
(zC~) is <<1.
Now, eq. (3), after the left-hand side has been differentiated, gives the (negative) growth rate for a single
particle and the resulting equation in rearranged
form is
G(R,n,t)=$=
- y+-CB)
(R>O)
P
(6)
the population
balance being given by
an@,
X, t) + ab(R.
at
X9t) G(R, X9t)i
aP
the right-hand side of this equation being set to zero
since there are no birth and death events within the
particle size range 0 - R,,. (Death, i.e. exit of particles from this size range occurs only at the boundary
R = 0.)
The initial and boundary conditions on the primary
set of equations, namely, eqs (l), (2) and (7) are given
by the following.
ICs (t = 0, all x):
c* = C*b = 0
(8)
c, = cab = c;
(9)
h(R, x, r) = hi,(R)
(10)
BCs (t > 0):
x=0,
x + co,
C*=Cj,
C” = C,& = 0,
R = L,,
ac,=o
ax
(11)
Ce = CBb = cj,
(12)
n(R, x, t) = 0.
(13)
The bulk concentration of A(CAa) is taken to be
zero since the reaction is fast, whereas the bulk concentration of B(Csb) is assumed to be that for
a saturated solution of B [for fine particles this is
a reasonable approximation; Sada et al. (1979)]. Reactant B is non-volatile as shown by the boundary
condition setting the concentration gradient (hence,
flux) at the gas-liquid interface (x = 0) to be zero. The
condition on the particle size distribution function
n(R, x, t) at R = R,,, implies that no particles are
entering the particle size range from “above”, i.e. sizes
greater than R,,,. (It may be noted that we have
consistently shown the function arguments for n by
writing it as n(R, x, t) . Rigorously speaking, the concentrations of A and B should also be written as
CA(x, t) and C,(x, t), respectively, but this has not
been done following the usual literature conventions.)
A solution of the above equations, along with the
relevant conditions, provides us with the main quantity of interest, namely, the specific rate of absorption
of A, which, from Higbie’s penetration theory, is
given by
RA = f
“RAi(t) dt
es 0
where, the instantaneous specific rate of absorption is
(15)
and t, = 4D,/(nki).
The other variables of interest are the concentration profiles, CA and Cg, the particle size distribution
function, n(R, x, t) and the moments thereof, such as
the particle number concentration,
R,.
= 0
(R > 0) (7)
N(x, t) =
n(R, x, t) dR
(16)
Gas absorption in reactive slurries
the average particle size,
Rmca”(x,
t)
=
so that n:(R) describes the initial state of the particles,
fed as a slurry, to the batch reactor at batch time,
St?-=
Rn(Rx,4 dR
(17)
N(x, 4
and, the local, volumetric ratio of solid to liquid,
&..
Qx, t) = f
R3n(R, x, t) dR
(18)
I0
which, for “low” values of 1, may also be taken to be
the local, fractional volumetric hold-up of the solid
reactant.
Bulk population balance
Since the bulk concentrations of A and B have been
taken to be fixed [conditions (8) and (9)], in the
formulation given above, the bulk material balances
for these species are not required. In the case of the
population balance, the bulk particle size distribution
function n,(R) [rigorously, nb(R, 6)] needs to be computed as a function of batch time, 0 (which is likely to
be much larger than the contact time represented by
tJ. The appropriate population balance on a batch
slurry reactor is
bn)(R,
0) =
465
e = 0.
The bulk analogs of quantities defined in eqs (16),
(17) and (18), and denoted by a subscript b, i.e. Nb(Q
R mcnn,b(e) and /b(e) may be obtained by replacing
n(R,x, t) in the above-mentioned
equations
by
n,(R, @, which in turn is obtained from the solution of
eq. (19) with condition (21).
Solution strategy
A semi-numerical strategy has been developed to
solve the species balances for A [eq. (l)], B [eq. (2)]
and the population balance [eq. (7)], along with the
appropriate conditions, as described below.
Consider first the population balance given by
eq. (7). Defining a new variable, o(R,x, t) =
n(R, x, Q/R, this equation may be recast in the form
(22)
where G(R, x, t) is the (negative) growth rate, dR/dt,
given by eq. (6). Now, from the method of characteristics, the left-hand side of eq. (22) may be taken to be
the total derivative, do/dt, which set to zero implies
that o(R, x, t) = constant = w(Ro, x, 0). Therefore,
x, t,) - n*(R, @)I dx ”
n(R,
R
c
-
&C(ndR,0) Gb(R,41
(19)
where the last term is on account of particle dissolution in the bulk phase. Since, the bulk liquid phase has
been taken to be saturated with E, the bulk (negative)
growth rate, Gb(R, g), given by
= n(Ro, & 0) _
Ro
= -
~~(c$-c,b)
,
oj
t23J
Rij = RZ +
2ShRDsM,
’
(Cf, - C,) dt’ (R > 0)
I0
(24)
so that this result substituted into eq. (23) finally gives
(20)
becomes zero for CBb = C$ and hence the last term in
eq. (19) may be dropped. We thus have a situation of
exclusive particle dissolution near the gas-liquid interface and none at all in the bulk. This is consistent with
the case of fast reaction (no bulk reaction) and fine
particles which keep the bulk phase saturated. The
integral term in the bulk population balance, eq. (19X
represents the exchange of material between the interface, composed of a mosaic of penetration elements,
and the bulk. While the first term within the integral is
the number (concentration) of particles in the differential size range R to R + dR that are being carried
awayfiom the interface by the departing penetration
elements, the second term gives the equivalent number
(concentration) of particles that are brought into the
“near-interface” zone from rhe bulk. A more detailed
discussion on this type of bulk balances may be found
in Mehra et al. (1988). Equation (19) requires an initial
condition which may be stated as
nb(R, f?) = n:(R)
fR
0
(R>O)
*
_
@= 0,
fib:01
the last equality being obtained by the use of condition (10). In order to eliminate the dummy variable,
Ro, we may now integrate eq. (6) with the initial
condition, R = R. at t = 0. This integration yields
PP
Gb(R, 0) = ;
(R>o)
$+G(R,x,t)$=O
(21)
n(R, x, 4
= n&/R’
+ (2ShRD,N,/pp)j;(Cs,
JR” + (2Sh,&%4,/pp)J~(C~
- CddflR
- C,) dt’
(25)
provided, R. d R,,. (else n(R, x, t) = 0), and where, it
should be recalled, that the dependence on x is
through the concentration, CP Thus, the particle distribution function, n(R, x, t), may be computed from
the initial distribution alone, i.e. rib(R)). The boundary
condition (13) makes no contribution to the solution
since n(R,&, x, t) is always zero.
If this final result, from the above eq. (25), is substituted into eq. (2) i.e. the species balance for B, we
have only the following equation:
a2cs acB
= at + zk2CACB - 4nD,ShR(C; - C,)
ax
DBz
s
R-R2n&h2 + (2&&&fw/P,)~&i
0
,/R2 + (2Sh,D,M&&,(Cs,
- cd dt’l
- CL,)dt’
(26)
dR
A. MEHRA
466
Table 1. Data sets used in this study
Standard set
Property
CZ
G
DA
DB
kz
It:.
a
hl
1:
ShR
LX
MW
PP
Sada et al. (1984)’
Value
Value
Units
3.00x 1o-2
9.00 x lo-2
2.00 x 1o-9
2.00 x 1o-9
1 x 10s
1.0
5.0 x 10-S
1 x lo2
0.1
0.1
1.0
1.0 or1 10.0
2.66 x lo-’
2.03 x lo-’
2.00 x 1o-9
1.64 x 10-9
1 x 10s
1.0
5.0x 1o-s
kmol/m”
kmol/m3
m’/s
m2js
m3/kmol s
14
1500
S
Z
5.0
1
m/s
m2/m3
nd
nd
nd
nd
2GO
kg;:ol
kg/m3
0.05n-aO.25
Note: nd = non-dimensional; na = not applicable.
t C02-Ca(OH)2 system.
and eq. (1) to be solved along with the initial conditions (B), (9) as well as boundary conditions (11) and
(12).
The above-mentioned
equations, in non-dimensionalform (see Appendix A), were solved numerically,
simultaneously, since these are coupled [eqs (1) and
(26)], by using an implicit, jinite difference discretization within an iterative procedure. This method
involves solving eq. (1) for CA(x, t) and eq. (26)
for CB(x, t) sequentially, using the last generated
values of the other variable. For instance, when solving eq. (1) for C,,, the last obtained solution for CB
was used. This iterative process was carried out
till convergence
was achieved. The values of
nb[ RZ + (2ShRDBM,/p,,) JA(C$ - CB) dt’] were obtained either by substitution into the expression for
n,(R) [such as eqs (37) and (3811 or by a cubic spline
interpolation on the discrete values of n*(R) stored in
tabular form (this is done only when solving along
batch time, 8, since only nb(R, 0) is known as an
analytical expression; nb(R, 0) is known only at discrete values of R). All the integrations with respect to
R were performed using Simpson’s rule; the integrals
with respect to time, t, such as, for evaluating
RA [eq. (14)] and that containing (C; - C,) as in
eq. (26) were evaluated with the help of the Trapezoidal rule.
The bulk population balance [eq. (19)] was solved
using the Euler’s method and the integrals in this
equation, with respect to x, were obtained from
Simpson’s rule. In this case, only nb at 0 = 0, i.e. n$!is
available as an expression; the subsequent values of
nb at 6 > 0 then become available only as a set of
tabulated numbers at discrete values of R.
More details of the solution procedure may be
found in Appendix A.
Table 1 lists the standard set of values for the input
parameters (physico-chemical properties and operating conditions) that were used for this study. The
proposed model was solved by varying one or two
parameters in this set, while keeping the others fixed
at the values shown.
RESULTS AND DEXXJSSION
In the first part of this study, the effects of changing
particle size(s) near the gas-liquid interface, on the
specific rate of absorption, have been considered.
Here, the variation of the specific rate with batch time
has not been examined and only the computation of
the specific rate with respect to the conditions prevailing in a penetration element has been considered. For
this purpose, a simple initial particle size distribution,
nb(R), was chosen, namely, an impulse function
n,(R) = No&R - R,,,)
(27)
which implies a monodisperse distribution, where all
particles are of size R,., at a number concentration,
No. This is thus the particle size distribution function
when the penetration element arrives at the gas-liquid
interface at time, t = 0. Substituting this into the last
term of eq. (26) and integrating with respect to R, this
dissolution term may now be written as
Rgd = 4nShR De(C; - C,)
*No
R&x - (2ShRD&w/p,)
J;(C;, - Cd
dt’.
(281
Apart from the simplicity of this initial distribution
function, the advantage of using eq. (28) is that numerical integration, in order to evaluate the integral in the
last term of eq. (26) with respect to R, is avoided. It
follows, then, from the definitions (16), (17) and (18)
that for this situation,
N&t) = No,
R,& Z (2Sh,D&,/p,)j;(CS
- C,)dt’
(29)
N(x, t) = 0,
R:,, < (2SW&Wp,)
J!&; - ‘3 dt’
(30)
Gas absorption in reactive slurries
467
-__ _ km
,
02-
x/(W.)‘”
and
t) = JR:,,
- (~SMM~,/P,)
MC;, - CB)dt’
(31)
which is equal to the size of all the individual particles
at that x and t, as well as
1(x, t) = y
+rn
pm
pm
l/2
Fig. 3. Variation of average particle size R,,,, and solid to
liquid volumetric ratio 1with distance from the gas-liquid
interface x at t = t, for different initial particle sizes R,,..
Data used: standard set; Initial particle size distribution:
impulse function.
smallest size of 1 pm-there
is indeed a particle-free
zone near the interface-but
for the larger sizes (5 and
10 ,um) the particles can be seen to have only reduced
in size and not dissolved completely. In fact, for the
largest size of 10 w the maximum change in size (at
x = 0) is of the order of 10%. For higher values of the
contact time, tc, this change may be expected to be
even more. Thus, for a given contact time, only those
particles larger than a certain size, can be assumed not
CR&,,
- (ZSh,D,M,/p,)I;(C$
;
x/m.)
Fig. 2. Variation of concentrations CA and Cs, average
particle size R,,., and solid to liquid volumetric ratio I with
distance from the gas-liquid interface x at t/tc = 0.1 and 1.0.
Data used: standard set with R_ = 1.0 pm; Initial particle
size distribution: impulse function.
L,.(x,
___.I
l:R,=l
2:R,=5
3:R_=lO
- C,)dt’13”.
(32)
The last two equations obviously hold only when
N(x, t) # 0. The value of No may be determined by
equating 1(x, 0), as evaluated from eq. (32), to the
desired value of lo, after choosing the value of R,,.
The resulting, typical profiles of the concentrations
of A and B, i.e. CA and Ce, respectively, the mean
particle size, R,,,,, and the solid to liquid ratio, 1, as
a function of the distance from the gas-liquid interface, x, at two different times, t, are shown in Fig. 2.
While the profiles of CA and C, are as expected for the
case of very fast/instantaneous
reactions (A and
B meet at a reaction plane, x = a), it may be observed
that the particle size decreases rapidly as one moves
towards the gas-liquid interface. In fact, very close to
the interface there are no particles at all - all the
particles in this zone have dissolved completely even in
less than 0.1 s! At the end of the contact period lasting
for time t = t, (= 1.0 s), the particle-free zone can be
seen to have “penetrated” considerably into the element. This kind of a profile for the particle size is to be
expected since the driving force, (C; - C,), is maximum closest to the interface and decreases away from
it as C, becomes significant (beyond the reaction
plane at x = A).
Figure 3 shows R,,,, and I, vs x at the time of
departure of the penetration element, i.e. at t = t,, for
different initial sizes of the particles, R,,.. The variation in these profiles with distance is steepest for the
to undergo any significant change in size and exist in
the “film” at a constant size. For the assumptions
made in this study, the condition for a particle to
undergo negligible change in size may be derived from
eq. (31), wherein substituting t = t, and replacing
R,,, by the initial size of any particle, R,,, we obtain
assuming the maximum driving force for dissolution
by setting Ce = 0. Thus, for instance, for the standard
data set listed in Table 1, particles with an initial
radius, Ro>>4 pm (say, about 5 times this, i.e. 20 pm)
will not undergo any significant change in size. Alternatively, a particle of R,, = 1 m will simply vanish in
about 0.06 s when subjected to the maximum driving
force for dissolution. The points at which the profiles
of R,,,, and 1 show a sudden change in slope (curves
2 and 3 of Fig. 3) correspond to the position of the
reaction plane because in the zone that contains negligible amounts of reagent B, the driving force for
dissolution is at its maximum and all particles in this
zone dissolve to the same extent (having started from
the same initial size)-hence the flat profile in this
region close to the gas-liquid interface.
The behaviour of the instantaneous
specific rate of
absorption of A, RAi, defined by eq. (15), as a function
of time, t, for “small” particles of 1 pm size is shown in
Fig. 4. It is clear from this that the rates computed
assuming a fixed, constant particle size are much
higher that those obtained from the realistic scenario
A. MEHRA
468
Fig. 4. Variation of instantaneous specific rate of absorption
RAl with time t from constant and changing size models, and
without particles. Also, variation of average particle size near
the gas-liquid interface R:..,, with t. Inset shows position of
reaction plane I with t for the three cases. Data used: standard set with R,, = 1 e; Initial particle size distribution:
impulse function.
where particles can change in size due to dissolution.
The variation of the average particle size at the interface, R,..,,(O, t) , denoted by R&, is also depicted in
this figure. The particles here dissolve completely
within 0.06 s. The inset shows the position of the
reaction front, 1, with respect to time, where A and
B meet to react instantaneously. The effect of the solid
reactant particles is to retard the rate of motion of this
plane by continuously supplying B to the liquid phase.
Therefore, this plane travels into the penetration element to the greatest extent when there are no particles
present. If particles of a constant size are assumed
then the extent of penetration is very small compared
to the case when the particle size is allowed to change,
since, in the former case, the particles continue to
supply the reactant B inexhaustibly, even though
these may have actually dissolved completely. Thus,
the actual supply of B to the liquid phase is much
less than what is assumed by models in the prior
literature.
Figure 5 shows a similar set of plots for “large”
particles of 10 pm. Here, the rates computed with
either the constant or changing size assumptions leads
to almost identical RAi vs t plots, since,the particles
are large enough for their size to be affected only
marginally in the given contact time. The maximum
variation in the particle size, for those located at the
gas-liquid interface, is about 10 %, as evidenced from
the plot of RZ,,, vs t in this figure. A similar observation holds for the reaction plane position, Iz, vs
t shown in the inset to this figure.
The (surface averaged) specific rate of absorption of
A, RA [eq. (14)], plotted as the enhancement factor,
dA, vs the initial particle size, R,,, from the constant
and changing particle size models is shown in Fig. 6.
The enhancement factor is defined as
4A= R.,IRp
Fig. 5. Variation of instantaneous specific rate of absorption
RAi with time t from constant and changing size models, and
without particles. Also, variation of average particle size near
the gas-liquid interface R&,, with t. Inset shows position of
reaction plane I with t for the three cases. Data used: standard set with R,,, = 10 pm; Initial particle size distribution:
impulse function.
a.0
010
R- Cd
Fig. 6. Variation in enhancement factor c$”vs initial particle
size R,,, from constant and changing size models at various
stoichiometric ratios q. Inset shows the asymptotic and
stoichiometric bounds on 4” vs 4. Data used: standard set;
Initial particle size distribution: impulse function.
where, R$., is the specific rate in the absence of the
fine particles. In this figure, the concentration ratio,
q = C&CA*), has been used as a parameter. For the
case of instantaneous
reactions, Ry
is given by
(Doraiswamy and Sharma, 1984)
/C$Cl
= J(E+
&?).
(35)
For the “small” particles, the difference between the
corresponding enhancement factor values from the
two models is very large. As the particle size increases,
the predictions tend to become closer and finally
coincide for “large” enough particles. An interesting
observation to be noted from this figure is that while
the enhancement factor curves for the constant particle size cases are closely bunched together for all the
Gas absorption in reactive slurries
I
/
/
/
/
7.3 -
2
_
/
/
/
/
I
/
1:ShR=O.
1
2:Sh.=l.O
3:Sh.=5.0
, ’
469
dissolve to an extent that will affect the rates significantly thereby lowering the enhancements vis-&is
the constant size model predictions.
Interestingly, the enhancement factor values from
the changing size model, in Fig. 7, for all the Sherwood numbers, tend to approach an identical plateau
at large contact times; the approach to this asymptotic value is faster for higher ShR. Coupled with the
discussion above, this implies that at this asymptotic
extreme the rate (or alternatively, enhancement factor)
becomes independent of the initial particle size as well
as the Sherwood number and depends only on the initial
solid loading, lW Also, if the modification of Uchida
et al. (1975) were to be taken into account, by using an
Fig. 7. Variation in enhancement factor 4A vs time of contact of penetration element t, from constant and changing
size models for different Sherwood numbers ShR. Data used:
standard set with R,,. = 1 pm; Initial particle size distribution: impulse function.
enhancement factor for the particle dissolution process, the overall benefit of this mechanism in further
intensifying the specific absorption rate is likely to be
moderated by quicker particle disappearance.
At this point it is also pertinent to look at the
stoichiometric
q values, these are clearly
well separated for the
changing particle size cases (in the smaller particle size
range). This happens because for the greater q values
(obtained by increasing the C; value) the rate of
dissolution is faster and hence particles become
smaller and/or disappear earlier resulting in the largest q case to have the lowest enhancement value.
Though the absolute value of RA increases with q the
corresponding increase in Ry is relatively more [according to eq. (35), RAbase increases linearly with q].
This causes a reduction in the 4A value. For the
constant size runs, the increase in RA and RF, with q,
is nearly the same - hence the bunching of these
plots. The most significant observation from this figure is that as the particle size decreases, the enhancements from the changing size model approach an
asymptotic plateau. The variation of this asymptotic
value with q is shown in the inset to Fig. 6. In other
words, even smaller particles will not increase the rate
of absorption, since these dissolve completely, anyway. This is in direct contrast to the predictions from
the constant size models.
As has been stated earlier, for particles to be “small”
enough for these to undergo significant change in size
due to dissolution, what matters is not just their initial
absolute size (and hence the dissolution rate) but also
the total time available for dissolution, i.e. the contact
time period, te. This is demonstrated in Fig. 7. For
large rates of dissolution (as represented by larger
values of the Sherwood number, Sh,J and/or for high
values of the contact time, t,, the divergence between
the enhancement factors predicted by the changing
and constant particle size approaches, respectively, is
substantial on account of the complete or partial (but
significant) dissolution of the particles. Thus, it may
be concluded that for higher values of t, (lower k,)
even “large” particles may become “small” enough to
dissolve substantially. Therefore, say, for the standard
data set shown in Table 1, if tc is increased to a few
seconds, even particles of 10 pm may be expected to
upper
limit on the r$A values,
as dis-
cussed by Yagi and Hikita (1987), as this is especially
relevant for “small” particles which tend to dissolve
completely within the contact time, tc.If all the material contained in the solid phase is deemed to be available “immediately” in dissolved form, a limiting value
of the enhancement factor may be written as
These values of 4A are also plotted in the inset to
Fig. 6. It may be seen that the values of e%,,predicted
from the model of Ramachandran and Sharma (1969)
can be made to exceed even this upper bound
by choosing a particle size which is “small” enough.
This is, however, stoichiometrically impossible. The
4” values computed from the proposed model are
indeed smaller than the bounds given by eq. (36). The
difference between the stoichiometric and asymptotic
limits is on account of the fact that no matter how
small the particles, there is always some point in the
penetration element beyond which the driving force is
insufficient to cause “immediate” dissolution, i.e. there
will always be some B “bound and immobilized” in the
form of particles. It should also be noted that rapid
particle dissolution may cause small, local convective
flows around a particle which may further enhance
dissolution rates. We have not attempted to estimate
these in our analysis in the present work, though these
could be incorporated simply by increasing the value
of ShR.
Figure 8 shows the effect of the initial (t = 0) solid
loading, lo, on the enhancement factor, 4A, for different initial sizes of the solid reactant particles. While
the difference in the enhancements from the two
models, for the smaller particles is large, it becomes
negligible for the largest particle size.
Uchida et al. (1978) have found that the absorption
rate did not depend upon the particle size, R, for
a fixed value of the solid-liquid interfacial area
A.
/
15.0
/
/
/
125
MEHRA
l:R,..=l.O +'m
2:R,,=5.0 /on
3:R,,=lO.O
pm
/
/
$
I
la0
Fig. 8. Variation in enhancement factor 4A vs initial
solid loading of particles I, from constant and changing size
Data
models for different initial particle sizes R,,.
used: standard set; Initial particle size distribution: impulse
function.
[A, = 3wt/@,R)] and deduced from this that kSidoes
not depend upon the size R. These workers used
a constant value of the ratio of weight fraction of solid
to particle size (wt/R) but varied both wt and R. An
alternative explanantion
is provided by the plots
given in Fig. 8. The 44 values at lo = 0.1,0.05 and 0.01
read, respectively, from the curves for 10, 5 and 1 pm
sizes give values between 2.1 and 2.4. This happens
because for the large particle size the rates are low
despite large loadings, whereas in the case of small
particles, these dissolve away completely and the
loadings are anyway low so that the rate may remain
substantially the same. Therefore, it cannot be concluded from these experiments that kSl is a constant
with respect to R. The above comparison is only to
demonstrate the effect of dissolution and size change;
the t&es of 4A mentioned above (read from Fig. 8)
do not correspond to the system of Uchida et al.
(1978).
The results discussed above have all been derived
using the impulse function, i.e. an initial (t = 0),
monodispersed particle size distribution. A logical
corollary to the observations made above is that for
a “wide” initial size distribution comprising “small”
and “large” particles, the smaller ones may dissolve
completely while the larger ones may remain substantially unchanged in size. This implies that the specific
rate of absorption is likely to depend strongly on the
width of the initial particle size distribution, for a fixed
initial average size. In order to examine this hypothe-
sis, the following initial particle size distribution
tions are used.
Distribution
func-
I:
n,(R) = Ci exp C- G(R,..
- RI1 (R d LA.
(37)
Distribution
II:
rib(R))= C1 exp (- Cz R)
(R < R,,3.
(38)
The values of the pre-exponential
parameter
Ci and the steepness parameter, C2 [a larger value of
this implies a narrower distribution clustered around
R = R,,, for eq. (37) and around R + 0 for eq. (3811
can be computed by fixing the maximum size, R,,,,
and setting the initial solid loading expression as well
as the average size expression, from eqs (18) [1(x, 0)]
and (17) [R,,.Jx,
0)], respectively, equal to the desired values.
Table 2 shows the effect of the width of the initial
particle distribution function on the enhancement factor, 4A, for the two mean sizes, Rmcan,o = 1 and 10 pm,
respectively, using the proposed theory. The enhancements (or alternatively, the specific absorption rate)
may be seen to be substantially affected by the width;
a broader distribution yields lower rates. Figure
9 shows the behaviour of the number concentration of
particles, N* = N(0, t) and the solid to liquid ratio,
I* = l(0, t), at the gas-liquid interface, as a function of
time, t. Figure 10 shows the corresponding variations
in the instantaneous specific rate, RAi, and the average
particle size, R&., = R,,,&O, t). (These figures are for
an initial, mean radius, R,,,cpn,o = 1 pm.) While the
number and volume of particles decreases with time,
the average size increases implying that the “fines”
have dissolved away leaving behind larger particles
which have not changed in size very significantly. For
an initial distribution of greater width the average size
increases by a larger amount, as can be noticed from
Fig. 10 as well as Table 2. Therefore, the average size
at any location x within the penetration element is
likely to have a maxima with time, t. For the
RL, curves shown in Fig. 10, this maxima is clearly
visible for the two intermediate values of R,., = 3 and
5 pm. Initially, the smaller particles dissolve faster
leaving the larger ones behind so that the average size
goes up while later even these particles start reducing
in size due to dissolution, thus decreasing the average
size. For the impulse function there are no small and
large particles; for the broadest distribution (curve
4 in Fig. 10) the reduction in size due to dissolution
has not yet overtaken the increase due to dissolution
of “fines”. From Table 2, it is interesting to observe
that, for Rmean,,, = 10 pm, as R,,, is increased to
make the distribution wider, the fraction of particles
that dissolve completely near the gas-liquid interface
in time t, increases (from 0 to 0.21) but the volume of
the solid that has dissolved decreases (from 0.25 to
0.12). This indicates that, upon widening the distribution, more “fines” are present in the system in order to
keep the initial average fixed and these dissolve completely. However, simultaneously, a larger volume of
the solid, which is contained in the larger particles
that are now present, now remains undissolved.
This trend is reversed for the smaller size of
R mean,0 = 1 pm. Now, all the particles for the case of
narrower distributions disappear. As the distribution
is widened, “coarser” particles which are present remain undissolved.
It has been pointed out in some studies (Sada
et al., 1981, 1984) that the specific rate of absorption
1000
500
100
10
1
10.0
11.0
12.0
16.0
19.5
20.0
= 2.16
725
970
loo0
62547
49,380
40,200
3.00
5.00
10.00
1.00
1.10
1.60
1.93
758
500
398
2387
2320
1548
1249
239
2.32
216
1.55
1.22
1.19
be’)
N,, x lo-‘s
N*
0.00
0.00
1.00
1.00
0.98
0.87
0.80
0.79
4.99
4.74
4.43
0.00
0.0089
0.015
Distribution II
5.18
5.18
5.14
5.11
2.20
218
2.11
1.86
1.70
1.68
Distribution I
N,
R&
0.00
1.60
2.74
0.00
0.00
0.00
0.00
0.91
0.91
0.92
1.02
1.10
1.12
R me.n.0
0.00
0.011
0.11
0.00
0.00
0.00
0.00
0.75
0.75
0.77
0.84
0.88
0.88
Note: constant = constant size model prediction; detailed information for last 3 rows shown in Figs 9 and 10.
t Time in which all particles at gas-liquid interface disappear.
10,000
1000
100
Impulse fn
2,322JKQ
194,500
71,490
R mun*o= 1.00 /ur& gy-“n’ = 21.0
Impulse fn
232
108
19.5
6.86
599
R mean,,,= 10.0 crm; &“‘““’
b-9
cs x 10-s
and P
at
tltc
0.0056
0.017
-
0;6
1.00
1.00
-
1.42
1.99
2.74
0.096
0.37
1.00
Standard data set
1.04
1.11
-
1.03
1.10
1.12
-
-
Standard data set
maxima
R mmzul.
cl
RZWUI
0.51
-
0.06
0.07
0.14
0.21
1
-
-
tltc
ALL’
Table 2. Effect of width of initial particle size distribution on enhancement factor +,, and solid-phase characteristics near the gas-liquid interface-N*, R&,,
412
A. MEHRA
(1984) have attempted to explain this by the use of an
ad hoc hypothesis that a thin zone, of width equal to
one particle diameter, near the gas-liquid interface
has “no particles suspending” in it. This lowers the
computed rate value since in the region closest to the
interface there are no particles to provide the rate
enhancing effect. In addition to the “fudge factor”
approach implied by this hypothesis, there seems to
be no physical reason as to why such a particle-free
zone should exist per se. It is also clearly against the
“spirit” of pseudo-homogeneous
models, where the
penetration element is viewed as a “pseudo-homogeneous continuum”, to invoke a physical discontinuity at some arbitrary point. An alternative explanation,
Fig. 9. Variation of number concentration of particles N*
and solid to liquid volumetric ratio I+, near gas-liquid interface, vs time t for initial particle size distributions of different
widths from changing size model. Data used: standard set;
Rmem.0 = 1.0 lun.
II
0.0
,,
0.0
I , ,,
I2
, ,,
0.2
1, ,,,‘,,
0.4
, , , ,
,I,1
0.1
01
,,
,o
.o
IO
t/t
Fig. 10. Variation of instantaneous specific rate of absorption RAI and average radius R&,, near the gas-liquid
interface with time t for initial particle size distributions of
different widths from changing size. model. Data used: standard set; Rmc.n.o= 1.0 pm.
obtained experimentally has been usually found to be
lower than that computed from the models proposed
by Ramachandran and Sharma (1969) or the equivalent unsteady one of Uchida et al. (1981). Sada et al.
therefore, may be derived from the theory proposed in
this study, in the light of the foregoing discussion; the
process of dissolution (partial or complete) of “small”
particles near the gas-liquid interface causes a lowering
of the amount of solid present in this zone leaving behind
larger particles, which in turn, is responsible for the
reduced absorption rates vis-&vis the earlier model
predictions. The results of calculations using the data
of Sada et al. (1984), for the carbon dioxide-calcium
hydroxide slurry system, are shown in Table 3. It can
be seen that the rates obtained depend strongly upon
the width of the initial particle size distribution for
a fixed, initial, average size and solid loading. As the
distribution is made wider, the enhancement factors
decrease and approach the experimental values reported by Sada et al. (1984). A more precise comparison than what is given in Table 3 is not possible since
Sada et al. (1984) have not reported the size distribution of the solids that they used. In fact, they have not
even reported the mean particle size but merely suggested that it may be about 10 pm (diameter). For the
computational
results shown in Table 3, we have
therefore used an initial radius of 5 pm. A slightly
different initial, mean size or a different distribution
may provide even lower calculated values for the
enhancement factors. It is important to note that Sada
et al. (1977a, 1984) have analysed their experimental
data by using a lumped parameter, kSIA,, and backcalculating the value of this by fitting the data to the
theory. The second study, mentioned above, attempts
a two-parameter fit by introducing an additional
Table 3. Comparison of enhancement factors from changing size model, for different widths of the initial particle
size distribution, with published experimental data of Sada et ~2. (1984); Rmcan,o= 5.00 pm
wt %
+
5
10
+
0.024
Distribution
Constant+
Impulse
I
I .
II
Experimental
10
0.051
R,
(ka4
5.00
8.75
10.00
15.00
5.00
1.5
0.080
20
0.114
25
0.152
N
4A
Iv,
1.60
1.73
1.44
1.37
1.23
1.1
‘Constant size model predictions.
2.22
2.32
1.85
1.74
1.49
1.3
2.78
2.85
2.27
2.08
1.74
1.5
3.31
3.35
2.59
2.41
1.99
1.6
3.82
3.83
2.94
2.73
2.24
1.9
1.00
0.89
0.85
0.78
-
RL.
RIIICP”,
0
0.95
1.06
1.09
1.18
I*
I,
0.87
0.94
0.94
0.97
-
Gas absorption in reactive slurries
3
._----_----
473
r-f 0
-0d
Q,
_ &e
:
I
2
1.h
\
2.1,
\
35-
50
100
2 h)
0 (5)
Fig. 11. Variation in enhancement factor c#J,,,
solid to liquid
volumetric ratio Ii,, number concentration Nb and average
particle size Rmcsn,,,, in the bulk, vs batch time 0. Data used:
standard set; Imtial particle size distribution I with
0
R mar = 1.1 pm and Rmean,&
= 1.0 pm.
factor, namely, the thickness of the “particle-free
zone” near the gas-liquid interface. This was probably
done in order to obtain a higher fitted value for the
lumped parameter k,,A, while still keeping the rates
low. The proposed model essentially indicates that the
“k&,” term cannot be treated as a constant and
decreases with time so that its “effective” value is
lower than that computed by using the initial values
of solid loading and particle size. Unlike the analysis
used by the previous workers, which is based on
“fitting”, the theoretical values shown in Table 3 do
not use any fitted parameter but require that the
particle size distribution information to be known
a priori for the rates to be truly predicted. The published data on sulfur dioxide absorption have not
been analysed in the current work since this system
has more than one instantaneous reaction and has
somewhat more complex features.
Up to now it is the behaviour of the point rate that
has been examined with respect to the initial particle
size distribution that characterizes a penetration element when it joins the gas-liquid interface. In the
second part of this study, the specific rate-batch time
trajectories for a typical slurry batch reactor have been
worked out (i.e. a sequence of point rates at different
times along the batch time scale). Generating this
sequence involves a solution of eq. (19) along with
conditon (21). Figure 11 shows the specific rate of
absorption, RA, and the bulk quantities-particle
number concentration, Nr,, solid loading, Ib and the
average size, Rmea+ as a function of batch time, 6,
using the proposed theory as well as the unchanging
particle size model for a starting average size,
R,?,,,,J = 1 pm.
While the constant size model [see Appendix B for
a simple Batch model based on Ramachandran and
Sharma (1969)] predicts complete consumption of all
particles in about 200 s, the changing size theory
shows that about 64% of the solid loaded into the
Fig. 12. Particle size distribution function Q(R) vs R at
different values of batch time 0. Other details as in Fig. 11.
2
Fig. 13. Variation in enhancement factor 4A, solid to liquid
volumetric ratio Ibr number concentration Nb and average
particle size R,,..,,,,, in the bulk, vs batch time 0. Data used:
standard set; Imtial particle size distribution I with
R mar = 11.0 pm and R&sn.b = 10.0 pm.
reactor is still unconsumed! This is consistent with the
fact that the specific rates computed from the constant
size models (plotted as 4” in Fig. 11) are much higher
than those from the changing size model. The corresponding evolution of the bulk particle size distribution
function is shown in Fig. 12 and the variation is as
expected. The area under these curves, representing
the particle number concentration, Nb, decreases as
0 increases while the mean radius, Rmea+ also decreases. Figures 13 and 14 depict analogous plots for
particles of a starting mean size of 10 pm. Here, the
predictions of the two models nearly coincide for quite
some time (approximately, 800 s) till the particle size(s)
become “small” enough so that particle dissolution
efiects near the gas-liquid interface cannot be ignored;
the predictions now cease to match and the constant size
assumption now overpredicts the specz$c rate value.
Interestingly, this implies that even for slurry reactors
which are initially loaded with “large” particles, there
will exist a time (on the batch time scale) beyond
A. MEHRA
474
R(w)
Fig. 14. Particle size distribution function n,(R) vs R at
different values of batch time 0. Other details as in Fig. 13.
which effects related to the “fineness” of the particles
cannot be ignored. Thus, the calculation of batch time
for a specified conversion of the solid is likely to be
grossly in error if particle size changes due to dissolution near the gas-liquid interface are not taken into
account.
CONCLUSIONS
It has been shown in this study that sparingly
soluble, fine reactant particles are likely to dissolve
significantly and undergo a substantial change in size,
near the gas-liquid interface, when slurries constituted by these particles are used to absorb a reactive gas.
This dissolution may occur, sometimes exclusively,
near the gas-liquid interface, i.e. within the penetration elements that make up this interface, provided
the reaction between the absorbing dissolved gas and
the sparingly soluble solid is fast enough to cause
the depletion of the dissolving solid species near this
interface.
The extent of particle dissolution depends on the
(initial) particle size distribution,
the Sherwood
number for the dissolution process, the concentration
driving force for the dissolution (which depends, in
turn, on the depletion of the dissolved solid species)
and the time of contact of the penetration element
with the gas phase (which is inversely related to the
liquid side mass transfer coefficient, i.e. the stirring
intensity).
The specific rate of absorption into slurries of fine
particles may be considerably overpredicted if the
particle dissolution effects near the gas-liquid interface are ignored. Thus, the values of the rate computed from the theory proposed in this study are
smaller than the corresponding values obtained from
prior models in the literature, which assume a constant, unchanging particle size near the interface.
However, for particles which are “large” enough (with
respect to the physicochemical properties and the
operating conditions) to allow for the neglect of these
dissolution effects the predictions from the proposed
model match closely with those from the previous
models. The current model therefore allows for establishing the limits of validity of the models already
available in the literature.
The models developed in this study have also been
used to demonstrate that the width of the initial
particle size distribution, that prevails within a penetration element upon its arrival at the gas-liquid
interface, has a considerable influence on the absorption rate because of the relatively faster dissolution of
the smaller particles.
Thus, the phenomenon of particle dissolution near
the gas-liquid interface may explain the commonly
encountered experimental finding that the measured
rates have usually been found to be lower than the
theoretically calculated values.
The evolution of the bulk-phase particle size
distributions for a batch slurry reactor have been
generated and it has been shown that the solid
reactant conversions computed from unchanging
size models may result in drastic underestimates
of the required batch times for specified conversion
levels.
Therefore, particle dissolution effects near the
gas-liquid interface are of tremendous importance
and cannot be neglected for slurry reactor analysis
when the particles in the slurry are in a fine state.
These considerations will be of significance even for
reactors starting out with “large” particles because the
reactor must pass through a state of “fineness” on its
way to complete conversion of the solid phase.
NOTATION
-a
CA
CAb
CB
G
CBb
Cl
c2
DA
DB
G
Gb
gas-liquid interfacial area per unit volume of slurry, m2/m3
specific solid-liquid
interfacial
area
C =3W(p,R)l, m2/m3
liquid-phase
concentration
of
A,
kmol/m3
solubility of A in liquid phase at gasliquid interface, kmol/m3
concentration of A in bulk liquid phase,
kmol/m3
liquid-phase
concentration
of
B,
kmol/m3
solubility of B in liquid phase, kmol/m3
concentration of B in bulk liquid phase,
equal to CS, in this study, kmol/m3
constant in eqs (37) and (38), me4
constant in eqs (37) and (38) m-’
diffusivity of A in liquid phase, m’/s
diffusivity of B in liquid phase, m2/s
same as G(R, x, t); growth rate for single
particle, m/s
same as G,(R, 6); growth rate for single
particle in bulk, m/s
second-order rate constant in liquid
phase for reaction between A and B,
m3/kmol s
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476
MEHRA
Sada, E., Kumazawa, H. and Lee, C. H., 1984, Chemical
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Sada, E., Kumazawa, H., Butt, M. A. and Sumi, T., 1977b,
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Uchida, S., Koide, K. and Shindo, M., 1975, Gas absorption
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APPENDIX A: NOTES ON NUMERICAL PROCEDURE
The primary eqs (1) and (26) may be written in nondimensional form as
-$=$+ K,ab
641)
and
Dz=-$+$ab-K,(l-b)
X
‘-
R’n,(,/R”
s cl
,/R2
+ K&(1
+ KS j;(l
- b)dT’)
- b) dT’
dR (A2)
respectively, where, a = C&x,
6 = C&i,
X = .x/a,
T = t/tc,
Kt = k2C&,
Kz = 4xD,Sh,,t,.
K3 =
2ShRD&t,M,/p,,
D = D,/D, and q = C;/(zCX).
The initial and boundary conditions (eqs (8), (9) and (1 l),
(12), respectively) then become as follows.
(ICS, T = 0):
X20,
a=ab=O
(A3)
b=b*=l
(A4)
(BCs, T > 0)
X=0,
X+co,
a=l,
a=ab=O,
ab
-=0
ax
VW
b=br,=l.
(A6)
The rates given by eqs (14) and (15) can be rearranged to
aive
(A7)
G-W
The above equations can be solved by using an iterative,
finite difference scheme which is implicit in time. Equations
(Al) and (A2) may be discretized by the use of a three-point
central difference for the space derivative which is implicit in
time. Each of these equations therefore yields a tridiagonal
system of linear equations, in the X domain, where the nonlinear terms such as, Klab, (K&J) ab and the last term of eq.
(A2) can be split or computed using the last available values
for the variable which is not being solved for from a given
equation. Thus, eq. (Al) may be solved for a using the last
computed values of b, followed by the solution of eq. (A2) for
b using the latest available values of a. This solution sequence
may be repeated till convergence, i.e. the values of a and b do
not change anymore. This iterative procedure is required to be
applied for obtaining a solution for a and b at T + dT
knowing the values at T, i.e. for every time increment.
The output values (such as, RAi, RA, N, R,,,, and I) for
every run were tested for their insensitivity to the value of
kz used, in order to ensure that the rate constant was large
enough for the reaction to be deemed instantaneous. The
condition for the reaction to be treated as instantaneous, in
the absence of reactant particles, is given by (Doraiswamy
and Sharma, 1984)
-
JKlW
(A9)
whereas, the analogous condition, in the presence ofmonodispersed particles, has been shown by Ramachandran and
Sharma (1969) to be
(AlO)
Condition (A9) is essentially required to be satisfied so that
depletion of species B occurs near the gas-liquid interface in
order to provide a driving force for particle dissolution to
occur in this zone. When particles are present, the reaction
rate (consumption rate of B) needs to be even higher for
depletion to occur since now the particles are also supplying
(extra) B. This is embodied in condition (AlO) and this was
satisfied for all the runs in this work. This condition is on the
conseruatioe side since the Ramachandran and Sharma
model assumes an inexhaustible supply of B.
A moving-boundary version of the model proposed in this
study was also developed where the concept of a reaction
plane, within the penetration element, moving with time is
used explicitly [as proposed by Uchida et al. (1981) for the
case of constant particle size]. This model yields results
which are within a few percent of those reported in this work.
However, this model is numerically not very stable since it
requires numerical differentiation.
The results obtained in this study were also ensured to be
free from any dependence on the various iteration tolerances
as well as the numerical parameters such as the step sizes on
X, T, 0 and the interval size of the table used to store values
of nb(R, 0) (this is relevant only for the cases where the cubic
spline interpolation is necessary).
It needs to be mentioned that R = 0 is strictly not an
admissible value since it implies that a particle does not exist.
Therefore, all integrals where R = 0 is given as one of the
limits should more rigorously be written with the limit
R = O+. Numerically, this problem was obviated by taking
a “very small” but finite value instead of 0, typically,
10e3 pm and it was ensured that the solutions were independent of this small but finite value.
A material balance check was made for the batch runs by
computing the amount of A absorbed till time 0 and comparing with the amount of solid reacted in as much time. This
check was satisfied to within 10%.
APPENDIX B: BATCH MODEL BASED ON
RAMACHANDRANAND SHARMA (V&9)
The rate expressions from the model of Ramachandran
and Sharma (1969) may be written as
Gas absorption in reactive slurries
and
477
conditions are
0 = 0, &caa.b= ha.
R~(Rmean.0)= 4 ,/4xI%h~Rmcan,oNo
Z
x coth(
+
where, Nz = 1:/(4/3)xRi.,.
(R4)
Integration of eq. (B3) gives
47rShRR~c.~,0N0@- d))
4~WWLR,,,.,oNo~
2
(B2)
where 6 = DA/kL. Here, RA may be obtained by first computing I by equating the right-hand sides of eqs (Bl) and
(B2) and substitution of this 1 value into either of the two
equations.
An overall reactor balance, relating the rate of absorption
of A to the rate of consumption of reactant B, is given by
which provides the relationship of Rme.n.b vs 6’(R’,,,...,b is
a dummy variable). The values of RA at any 0 may be found
by substituting the appropriate value of Rmc.n,b into eqs (Bl)
or (B2). This formulation implies that
Nb(B) = Nb”
lb(e)
>
=
zRA(Rms.n.&
(B3)
For a monodisperse, initial particle size distribution,
where all particles are of starting size R,,,, and assuming
that all the A that has been absorbed till time 0 has consumed
all the particles to an equal extent, the appropriate initial
=
(B6)
Nw3)nR:e,,,b.
In the context of the constant size models, the predictions
using the above formulation and those obtained from the
unsteady-state variant, as given by Uchida et al. (1981), are
nearly the same and also match the values that may be
computed from the model proposed in this study by deliberately holding the particle size constant.