10 1 1 1017 6199 PDF
10 1 1 1017 6199 PDF
10 1 1 1017 6199 PDF
ABSTRACT
Crundwell, EK. and Bryson, A.W., 1992. The modelling of paniculate leaching reactors--the popu-
lation balance approach. In: W.C. Cooper and D.B. Dreisinger (Editors), Hydrometallurgy, Theory
and Practice. Proceedings of the Ernest Peters International Symposium. Hydrometallurgy: 29: 275-
295.
A mathematical model of leaching reactors using the population balance is presented. The model
incorporates the material balances for the reactants and products, the population balance which de-
scribes the change in particle-size distribution, and the energy balance. The effects of the particle
kinetics, the particle-size distribution, and the residence-time distribution are accounted for.
The approach is discussed for the design of multi-stage, continuous reactors, and examples of the
application of the model are given for various forms of the leaching reactions.
The method is then used to develop an a priori model for the pressure leaching of sphalerite, and
the results are compared with those of the Cominco's plant at Trail, British Colombia, Canada. The
model incorporates the leaching reactions, the oxidation of ferrous sulphate, the precipitation of
plumbojarosite, and dissolution of oxygen from the gas phase into the solution. The kinetic parame-
ters for each of these reactions have been obtained from independent studies. The model predicts
results that compare favourably with data from the plant operation.
INTRODUCTION
Bartlett [ 1 ] has presented models for continuous leaching in both tank and
plug-flow reactors based on residence-time distributions. Implicit in his treat-
ment is the assumption that the particle kinetics are linear (i.e., the rate of
shrinkage is independent of concentration). A similar assumption was made
by Ruether [2 ].
The optimisation of reactor configurations for leaching was discussed by
Henein and Biegler [ 3 ]. They considered both plug-flow and continuous
stirred tank reactor (CSTR) reactors, and incorporated leaching kinetics of
monosize particles. Papangelakis et al. [ 4 ] on the other hand considered the
leaching of an arsenopyrite ore and, because the rate of reaction is controlled
by the surface reaction, they were able to use an approach similar to that of
Ruether [2 ] to model the CSTR reactor.
However, in order to model the leaching of particles of different sizes in a
reactor where the reaction is controlled by product-layer diffusion film trans-
fer, or by mixed control, it is necessary to solve the population balance equa-
tion to determine the outlet size distribution. Such an approach was adopted
by Sepulveda and Herbst [ 5 ]. They solved the population balance equation
for the various controlling mechanisms, and for various configurations of tank
reactors.
The approach of using the population balance to determine the change in
particle-size distribution during leaching is further examined here. The ef-
fects of different reaction mechanisms, various reaction orders, and particle-
size distributions on the reactor performance are examined, particularly for
the case where there is stoichiometrically sufficient reactant in the feed.
The population balance is then combined with the material and energy bal-
ances for a number of components in order to model the performance of the
zinc pressure leaching reactor at Trail, B.C. The model developed is an a priori
model, in the sense that the rate constants have all been determined indepen-
dently, so that there are no adjustable parameters in the model. A sensitivity
analysis for the various rate constants is also performed.
Kinetics of leaching
The leaching of liberated mineral particles by a reactant in solution is a
heterogeneous reaction that frequently results in the formation of solid reac-
tion products which form on the unreacted core. This can be represented by
the generalized reaction:
aA(s)+bB(aq)~cC(s)+dD(aq) (l)
The kinetics ofleaching are often described by the unreacted shrinking-core
model. This model envisages a boundary at the surface of the unreacted core
MODELLING OF PARTICULATE LEACHING REACTORS 277
that moves towards the centre of the particle and a layer of porous solid prod-
uct that forms on the unreacted core as the reaction proceeds. The leaching
reaction involves the mass transport of reactants to and products from the
mineral surface, and involves the reaction at the unreacted surface. One or
more of these steps may be rate-controlling.
For the purposes of modelling leaching reactors it is necessary to know the
rate of shrinkage of the size of the unreacted core, and the time for complete
conversion of the particles. The rate of shrinkage, R (I,L), of the size of the
unreacted core, l, can be described by the shrinking-core model [6] and is
given by Sepulveda and Herbst [ 5 ] as:
dl
R,(I,L)=~-[ l I
PA k - ~ + 2 D a C ~
-2(a/b)
I
(l-12/L)+k-~n(l/L)2
] (2)
where:
PA= the molar density of the mineral;
ks = the surface reaction rate constant;
Da = the effective diffusion coefficient ofthe reactant B in the porous product
layer;
Ca = the reactant concentration in the leaching reactor;
L = the initial particle size;
kr= the film transfer coefficient.
The first term of the denominator on the right-hand side of eq. (2) de-
scribes the contribution to the rate of shrinkage of the unreacted core by the
rate of the surface reaction, while the second and third terms of the denomi-
nator describe the diffusion of the reactants through the product layer, and
through the film layer, respectively. If the first term of the denominator is
larger than the others, the reaction is said to be controlled by the surface re-
action, if the second dominates, the reaction is controlled by product-layer
diffusion, and if the third dominates, the reaction is controlled by film trans-
fer. Of course, any combination of these processes may also control the reac-
tion, in which case the reaction is said to be under mixed-reaction control.
It is important to note that the rate of shrinkage is dependent on both the
initial particle size and on the unreacted core size if the reaction is product-
layer diffusion controlled or film transfer controlled. This means that the rate
of shrinkage is not linear, and that in order to calculate the conversion cor-
rectly, the population balance for the system has to be solved.
The time required for a particle of original size L to dissolve to such an
extent so as to have an unreacted-core of size l is given by [ 5 ]:
p,, [,_._, ),.,_,31
T(I,L) =2(a/b)Ca -~ + 12Da -3/2+L2 + ~f-L-5_! (3)
278 F.K. CRUNDWELL AND A.W. BRYSON
where:
Rj = the growth rate of the unreacted-core size;
RL = the growth rate of the total particle size;
Q= the volumetric flowrate of slurry to the reactor;
V= the volume of the reactor.
The subscripts o and 1 = the inlet and outlet conditions, respectively.
If the particles retain their original size, then RL = 0, and if the volumetric
flowrate of slurry through the reactor can be regarded as being constant, then
(2o- Qt, and the population balance becomes:
O(RInt)
+ v 01 (5)
with the boundary condition that R~nt-,0 as I--,oo. The population balance
differential equation can be written in dimensionless form by using the fol-
lowing dimensionless variables:
where: T * = l/R?dl*
From eq. (2) the rate of shrinkage can be written in dimensionless form as:
-c~
( (8)
3t~r) \L*J + 12t~,LD /*-- -~-~)+tsR
* .
where:
t~n = [,/(3krfl)
t~,LD -~ E 2 / ( 1 2 D a f t )
tlR=E/(k~#)
fl=L/ ( 3kr) +L2/(12DB) +L/ks
By definition, t~o + t~,LD+ tla = I. The significance of these four parame-
ters is that they express all the information on the kinetics of leaching in di-
mensionless form. t~o is the contribution to the time for complete dissolution
of the average size particle in the feed due to external film diffusion, t~LD is
that contribution due to product-layer diffusion, and tlR is that contribution
due to surface reaction.
The distribution of unreacted-core sizes is given by:
Llla~
nT(i*) = al, f nT(l*,L*)dL* (9)
From the number density distribution given by eq. (9) the conversion can
be calculated from:
fot'~a"l*3nT(1")dl*
x=l (~0)
f ~'~ l*3n, ( l, )dl,
where:
a,j= the stoichiometric coefficient of the ith component participating in the
jth reaction;
Fro= the molar flowrate of the ith component leaving the reactor;
¢j= the extent of the jth reaction.
a~j is positive for products and negative for reactents. The molar flowrate
can also be expressed in terms of the conversion of the ith component, X,:
F,,--F,o(I-X,) (13)
The energy balance for a steady-state adiabatic flow process is given by:
tlo=[t, (14)
where/:/o and/:/~ are the enthalpy content per unit time of the inlet and outlet
streams, respectively, for this process. The enthalpy is given by:
where d/t~ and ~p; are the heat of formation at 298 K and the molar heat
capacity for each of the components. Assuming the molar heat capacities are
approximately constant between the temperatures of the inlet and outlet
streams, and substituting eqs. (12) and (13) for a single reaction and eq.
( 15 ) into the energy balance, eq. (14) gives:
MODELLING OF PARTICULATE LEACHING REACTORS 281
~ E o Cp,( T, - 7 o ) = ~ ( - A H R ) = F j X j ( - J / t R ) (16)
i
where:
W ,~
Single tank
Figures 1 and 2 illustrate the results for the model for an isothermal reactor
for the limits of rate control by surface reaction (t~R = l ), by product-layer
diffusion ( IpLD
* = 1 ), and film transfer (t~o = 1 ). The feed distribution is the
gamma distribution, which in dimensionless form is given by:
1.00- -
0.85- -
C
0
. m
C_
0.70 -
>
C
0
FirsL-order reacLion
0.55--
~e= 0
SurFace reecLion
...... ProducL-lauer dirr
........... Film Lransrer
o.#o I I I I I I I
0.00 O.TS I .50 2.25 3.00 3.75 4.50 5.25
O
Res,dence Lime, T
Fig. 1. Effect of the controlling mechanism on the conversion for excess of reactant. Feed distri-
bution: Gamma, a*:=0.5.
0.80=
0.?0=
C
0
01
L
~1
> 0.60-
t-
O
0,50-
............. ..........
PPoducL
2. - I a~ler
°;o
d i FF
........... Fi Im LrensFer
040 I I I I I I I
0.00 0,75 1.50 2.25 3.00 3.75 4.50 5.25
Residence Lime,
Fig. 2. Effect of the controlling mechanism on the conversion for stoichiometric amount of
reactant. Feed distribution: Gamma, a .2 = 0.5,
MODELLING OF PARTICULATE LEACHING REACTORS 283
is shown in Fig. 3 for the case in which chemical reaction is the rate-control-
ling mechanism (t*SR= 1 ). In this case, the rate of shrinkage is given by:
R 'f-CT,m (18)
T(O,L) 2(a/b) -
(19)
0.90-~
.....................--''''''''''' " ,-, .
0.80- Go w
..o o o
C
0 #.#so~'°°
, n
0.70- ## ............
01 • ... ........
t ¢ ......
# .... ..
# . ....'"
> #.# .... .... ..'
#. ....."
t- I ....'"
O 0.60-
0
0.40 I I I I I I ,'
0.00 0.75 I .SO 2.25 3.00 3.75 4.50 5.25
O
Residence time, "c
Fig. 3. Effect o f reaction order on the conversion for a stoichiometric amount o f reactant. Feed
distribution: G a m m a , a .2 = 0.5.
284 F.K. CRUNDWELL AND A.W. BRYSON
0.80-
C
0 0.75-
. _
(/)
f..
>
t- 0 . 7 0 -
0
0
0.65-
0.60 I I I I I I I
0.00 O.?S I • 50 2.25 3.00 3.75 4.50 5.25
Fig. 4. Effect of the variance of the feed distribution (gamma) on the conversion for stoichio-
metric amount of reactant.
! .00 -
~ ank
ank
#
3
0.90
Tank 2
C
0
, u
U)
LOS0-
rank ]
>
C
0
(.)
0.70-
Pnoduct. laden
i f'Fus i on
o= ]
0.60
', I I I I I I I
0.00 0.50 ! .00 I .50 2.00 2.50 3.00 3.50 4.00
$
Residence time,
Fig. 5. Co-current operation of leach reactors for stoichiometric amount of reactant. Feed dis-
tribution: Gamma, a -=0.5.
MODELLING OF PARTICULATE LEACHING REACTORS 285
The zinc pressure leach plant at Trail, B.C., has been in operation since
1981 [ 8 ]. The four compartment autoclave has a working volume of 100 m 3,
and the mean residence time at the design feed rate is about 100 min.
The concentrate is fed as a slurry of between 68 and 70% solids ground to
98% less than 44/~m. The sphalerite concentrate from the Sullivan Mine has
the following composition: 48.6% Zn, 5.7% Pb, I 1.6% Fe and 32% S. The
solution to the autoclave contains 50 g/l Zn and 165 g/l H2SO4.
The autoclave is operated under a total pressure of 1300 kPa and the oxy-
gen partial pressure is 750 kPa. Oxygen is supplied by spargers to the first
three compartments, and each compartment is agitated very well. The feed
acid has a temperature of 70°C and the heat of reaction raises the tempera-
ture in the reactor to the operating temperature of about 150°C. Plumbo-
jarosite precipitates in the second, third and fourth compartments and con-
tains 2% Zn [ 9 ].
The reactions that occur in the autoclave are the leaching of sphalerite by
ferric sulphate, the precipitation of plumbojarosite, the oxidation of ferrous
sulphate to ferric sulphate by dissolved oxygen, the leaching of iron sulphide
from the concentrate, and the dissolution of gaseous oxygen. These five reac-
tions can be written as:
Model description
Kinetics of reactions
The kinetics of the dissolution of iron-containing sphalerite concentrates in
ferric sulphate and ferric chloride solutions have received a large amount of
attention [ l 0-14 ]. The reaction mechanism is thought to be electrochemical
in nature, and the iron content of the sphalerite increases the rate constant of
the surface reaction linearly [ 13,14 ]. Although there are no reported studies
on the rate of reaction of the Sullivan concentrate, it is assumed that this
concentrate will react at a similar rate to other concentrates with similar im-
purity levels of iron.
The concentrate from the Gamsberg Mine in South Africa contains 9.08%
iron, while that of the Sullivan Mine contains 11.6% iron. The Gamsberg
sphalerite concentrate has been studied extensively [ l l - 13 ] at various ferric
and ferrous sulphate concentrations and at temperatures between 60°C and
90 ° C. The kinetics of dissolution of this concentrate can be described by a
surface reaction of order one-half (i.e., m = ½), with a rate constant of
k's=6190 e x p ( - 5 5 3 3 / T ) (mol/ma) ~/2 m/rain and an effective diffusion
coefficient for the diffusion of ferric ion in the product layer of
Da= 5.46× l0 -7 e x p ( - 2 8 8 7 / T ) m2/min. The expression for the rate of re-
action when the surface reaction is controlling has been linearised by minim-
ising the integral of the sum of squared errors in the range of interest. In this
case the linearisation has been performed in the region of 3-10 g/l Fe 3+, which
results in a linear rate constant of ks= 542.3 exp ( - 5533/T) m/min.
The rate of shrinkage is then given by:
R~'- -C~[7.9X 10~Sexp(2887/T)+ 1.778X 10-3exp(5533/T)] (20)
4.74X lO-4exp(2887/T)(l*-I*"~
L*] + 1.778X 10-3exp(5533/T)
where C[ =CB/128.9. The time for complete conversion of the average par-
ticle size (22.76× l0 -6 m) at a ferric ion concentration of 128.9 mol/m 3 is:
i
K= 10--36°8 [ Fe3 + ]
- [H +]2 m3/m°l (22)
Process parameters
The molar flow rates of the feed were obtained from Martin and Jankola
[8 ]. Dreisinger and Peters [ 16 ] provided data on the feed size distribution
(on a mass basis). This distribution is described by a Rosin-Rammler distri-
bution with a mean size of 22.76 × 10 -6 m, and a slope parameter of 1.563.
288 F.K. CRUNDWELL AND A.W. BRYSON
(27)
where Qsol is the volumetric flowrate of solution. The extent of reaction 3 can
be calculated from the material balance for the ferrous ion species in solution.
The extent of reaction 3 can be evaluated from the equation:
O=e~_ksV(F7o_2Ci-¢3-3¢4)2(F,o-l/4e~+es)Q~o~ (28)
The extent of reaction 5 may be calculated from the material balance for
oxygen:
TABLE I
I ZnS Ft = F t o - e t
2 0 : (aq) /:2 = F , o - !/4e~ + es
3 H20 F 3 - F3o- 2e2 + ½e3
4 H+ I"4=/'~o + 2e2- ~3
5 SO~- F5 =/:so- ~e2
6 Fe -~+ F6= I % o - 2 e l - c 2 + % - 2e4
7 Fe 2+ FT~-FTo+2el -c3+3e4
8 Jarosite Fa= Fso+ !/3~2
9S lq~= lq~o+ct +~4
10 Zn "+ Fio= FIoo+el- 0.058e2
! I FeS FI i = Fi io- ~4
MODELLING OF PARTICULATE LEACHING REACTORS 289
The operating temperature of the autoclave may be obtained from the en-
ergy balance. Since a temperature profile between the different compartments
is not reported, it is assumed that, within the reactor, there is significant heat
transfer between the compartments but that the reactor a- 'Jhole is an adi-
abatic reactor. The adiabatic energy balance is given by:
FtotCp ( T - To) = e! ( - A/~°! ) + E3( - - zJI'~IO3 ) "~"E4( - - ZJ/'~rO4 ) (31)
where:
Ftot = the total molar flowrate to the autoclave;
T= the temperature within the reactor (equivalent in each compartment);
-AHR! =the heat of reaction for reaction 1 ( - 29.8 kJ/mol);
-zJHR3=the heat of reaction for reaction 3 ( - 102 kJ/mol);
-ZfHR4=the heat of reaction for reaction 4 ( -80.6 kJ/mol).
The specific heat capacity, Cp is average value for the inlet and, since the
molar flowrate is dominated by water, this is taken to be that of water.
The left-hand side of eq. (31 ) represents the rate of energy removal from
the reactor, and will be denoted R (T). R (T) is a straight line as a function of
the reactor temperature. The right-hand side of eq. (31 ) represents the rate
of energy generation due to reaction, and will be denoted G (T).
Model solution
Equations (25)-(31 ) express the material balances for each of the com-
ponents entering and leaving each compartment in the reactor and the overall
energy balance. The rate processes for each of these reactions have been de-
termined independently of this investigation and, therefore, this is an a priori
model of the reactor, that is, there are no adjustable parameters in the model.
The solution of eq. (25) involves solving the population balance to obtain
the conversion. Under the conditions of the Trail operation, it is expected
that the dissolution mechanism is controlled by both surface reaction and
product-layer diffusion, with product-layer diffusion dominating, that is,
t~,LD = 0.78.
The extents for reactions 1-5 may then be obtained at a particular temper-
ature from the conversion of the sphalerite particles, and the material balance
and equilibrium equations represented by eqs. (25)-( 30 ).
290 F.K. CRUNDWELL AND A.W. BRYSON
Model results
Martin and Jankola [ 8 ] provided operating data for the leach autoclave at
two different feed flow rates, and also noted that the autoclave could operate
at up 200% above its design capacity. Figure 6 represents their data, and illus-
trates the prediction by the model for a temperature of 150°C for various
flowrates of the feed slurry to the reactor on the reactor performance.
This figure indicates that the model is a good representation of plant data
that are available. The model predicts a lower conversion than that obtained
on the plant for the first two compartments, and a higher conversion than that
obtained in the last two compartments. Clearly, further data need to be ob-
tained in order to thoroughly test the model.
The solution of the energy balance, eq. (31), is illustrated in Fig. 7. The
model predicts an operating temperature of 140 ° C. The plant operates at be-
tween 145 ° C and 155 oC.
The details of the molar flow rates within the reactor are given in Table 2.
The discharge from the fourth compartment can be compared with that ob-
tained in the plant. The model results are in good agreement with the plant
data. It is interesting to note that the model predicts that there is no precipi-
tation ofjarosite in the first compartment of the reactor, which is observed in
practice. In addition, the model predicts that most of the consumption of oxy-
gen (es) occurs in the first compartment of the reactor. Very little oxygen is
consumed in the final compartment, and for this reason oxygen is not sparged
1 ,OO -'~
CompaPLmenL
A 4
C 0,90- 2
0
(/)
t.
(P
>
E
0
o 0.80
Model
Plane
0,70 -I I I I I
0.50 o.:,s 1.oo 1.2s 1.so 1.75
C) RCT3
,e--,
32.0~
X
C G(T)
E 24.0 1
--)
t-~ 1 6 . 0 ~
I--
f_
o 8.00--
rv" 0.00 I I I I I I I I
90.0 100 1I0 120 130 1#0 150 160 170
T e m p e r , at. or, e, °C
Fig. 7. The solution of the energy balance for the Trail autoclave.
TABLE 2
IThis refers to the total plant consumption given by Martin and Jankola [ 8 ].
into the last compartment. The actual and calculated conversions for plant
operation at 105% of design capacity are given in Table 3.
Although this is an a priori model, it is instructive to examine the effect of
the kinetic parameters on the reactor performance. The kinetics of the leach
reaction are incorporated in the time for complete conversion of the average
particle size. Figure 8 shows that, as expected, this parameter has a strong
influence on the reactor performance. The kinetics of the leaching reaction
292 F.K. C R U N D W E L L A N D A.W. B R Y S O N
TABLE 3
Compartment
1 2 3 4
% .00 - CompaPLmenL
4
3
0.9S--
r"
0
O9 0 , 9 0 -
C.
C
0 0,8S-
0,80-
Model
$ Plant
O. ?S Li |* - - ] i i
~o.o 2o.o 30.0 40.0 so.o
T¢O,Lav el, min
Fig. 8. The effect of the time for complete conversion of the average particle size on the conver-
sion for the Trail autoclave.
that have been used in the model have been extrapolated from data obtained
at temperatures of between 60°C and 90°C. Since this is such an influential
parameter, it is suggested that the kinetics of this reaction be examined in
detail, especially in the high conversion ( > 95%) region which is applicable
to the fourth compartment.
The effect of the rate constant for the ferrous sulphate oxidation reaction is
shown in Fig. 9. This figure illustrates that at a rate constant greater than
0.5 X l0 -3 m6/(moF min) this parameter has very little influence on the re-
actor performance. Such low rate constants are not expected.
The effect of the rate constant for oxygen dissolution is illustrated in Fig.
10, which shows that this parameter can have a strong influence on the reac-
tor performance if it is below 5 min-~. This would mean that the compart-
MODELLING OF PARTICULATE LEACHING REACTORS 293
Compartment
1 .00- .----" 4
v
...-.3
0.90- "" 2
£ 0.80-
0
¢. 0 . 7 0 -
>
r" 0 . 6 0 -
0
0
0.50-
0.40-- * P I ant
Model
0.30 I I I I I I I I
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
Fig. 9. The effect of the rate constant for ferrous sulphate oxidation on the conversion for the
Trail autoclave.
Compartment
I.00 --
, .
'"" 2
t- ........... 1
O
'" 0.80-
(n
£.
>
r"
0
(..1
0.60 -
Model
P I ant.
0.40 I I I I I I I I
0.00 5.00 ,0.0 ,s.o 20.0 2s.o 30.0 35.0 40.0
-1
k S , min
Fig. 10. The effect of the rate constant for oxygen dissolution from the gas phase to the solution
on the conversion for the Trail autoclave.
ments were very poorly agitated, which is not the case, or that the gas sparging
into the contents of the compartment was very inefficient. Since this has such
a strong influence on reactor performance and is relatively easy to rectify, it
is unlikely that the reactor would be allowed to operate under these condi-
294 F.K. CRUNDWELL AND A.W. BRYSON
CONCLUSIONS
The population balance has been used to describe the change in particle size
distribution for the leaching of a mineral in a continuous stirred tank reactor.
The effect of various operating parameters, such as the feed size distribution
and the reactant concentration, and various kinetic parameters, such as the
reaction order, and the reaction mechanism have been examined.
This model has been applied to the modelling of the zinc pressure leach
reactor at Trail, British Colombia, using independently determined kinetic
parameters. The model and the plant data are in good agreement for conver-
sion of sphalerite, for the consumption of oxygen and for the production of
jarosite. In order to produce a model that is numerically more accurate, it is
important to have a better understanding of the co-precipitation of zinc with
the plumbojarosite, and to have experimental data for the kinetics ofleaching
at the conditions of the reactor, especially in the very high conversion region.
NOMENCLATURE
0* 2 Dimensionless variance
T Residence time in reactor, min
Stoichiometric ratio (see eqn. ( 11 ) )
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