A Mathematical Model for Heap Leaching of One or More Solid Reactants From Porous Ore Pellets

A Mathematical Model for Heap Leaching of One or More Solid Reactants from Porous Ore Pellets

DAVID G. DIXON and JAMES L. HENDRIX

A mathematical model is derived in dimensionless form for heap leaching of one or more solid reactants from nonreactive, porous, spherical ore particles. The model is for the interpretation of column and heap leaching data and for use in the design and scaleup of heap leaching processes. Data from experimental column leaching tests are presented which validate the model. The scope of the present study is limited to the case where the rate-controlling reagent is a component of the lixiviant solution only and not of the gas phase. The effects of particle-scale kinetic factors, heap scale and operating variables, competition between multiple solid reactants, and particle size distribution are examined using the concept of the heap effectiveness factor. It is shown that heaps operate either "homogeneously," with distribution of reagent throughout the heap at all times, or in a "zone-wise" manner, depending mostly on a single parameter. The observed value of this parameter deviates from the predicted value in inverse proportion to the degree of contact effectiveness between the lixiviant solution and the ore particles. A rough correlation between the contact effectiveness and Reynolds number is generated from the simulation of column test results.

I. INTRODUCTION

H E A P and dump leaching have become the dominant modes of treatment for low-grade ores in recent years, most notably of gold, silver, and copper. The method is simple, economical, and adaptable to any sized operation.

Though heap leaching has been a boon to the mining industry, the process still suffers from its share of problems and uncertainties. In many cases, extraction of metal values does not fulfill expectations. Also, many operators are unable to make sound design decisions or solve problems which may arise due to insufficient in- formation about the physical processes occurring within heaps. In addition, rinsing reagents out of spent ore to comply with environmental regulations can be costly and time-consuming, often necessitating long-term remedial action.

If heap leaching is to have continued success in the future, not only as a method for leaching low-grade ores but for other purposes as well, a more sophisticated approach to design must be developed. The successful de- velopment of innovative heap leaching technologies, such as the rapid biological pretreatment of refrac- tory gold ores, and of better ways to operate and de- commission existing heaps and dumps will require a basis in sound engineering principles.

Several mathematical models have been developed for large-scale coarse ore leaching processes, including heap and dump leaching, as well as in situ solution mining. Roman et al.lq and more recently Chae and

DAVID G. DIXON, formerly Doctoral Candidate with the Department of Chemical and Metallurgical Engineering, Mackay School of Mines, is now Assistant Professor of Metallurgical Engineering, Department of Metals and Materials Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z4. JAMES L. HENDRIX, Dean of Mackay School of Mines and Professor of Chemical Engineering, is with the University of Nevada, Reno, NV 89557.

Manuscript submitted January 11, 1993.

Wadsworth L21 have developed models for large scale leaching of copper oxides. Both of these models are for the one-dimensional (l-D) isothermal case and assume pseudo-steady-state, "shrinking-core" type kinetics at the particle level.

Notable models of copper sulfide dump leaching processes include the work of Harris I3l and Cathles and co-workers, t4.51 The in situ solution mining of copper sulfides has recently been modeled by Gao et al . [6'71 With the exception of Harris, these models all include oxygen and heat balances in one or two dimensions and assume particle-level kinetics of the modified shrinking-core or "reaction-zone" type, as derived by Braun and co- workers. 18J Bartlett m has derived a particle leaching model for copper sulfides which does not assume steady state, but to our knowledge it has not been incorporated into a model of a large-scale leaching operation.

Much less attention has been focused on the modeling of precious metals heap leaching. To the authors' knowledge, only one previous attempt has been published to date, that of Prosser. l~~ His model assumes cyanide ion as the rate-controlling reagent and shrinking-core type kinetics as outlined by Box and Prosser. I~zJ

In this study, a general model is derived for applica- tion to the heap leaching of any porous, low-grade ore when the rate-controlling reagent is strictly in the aqueous phase. Incorporated is a general model, fully developed in a previous article, L~21 for solid reactant dissolution and diffusion in porous, spherical ore particles which does not assume steady state. This particular model then provides the rate term for a global heap-scale model based on the l-D, plug-flow reactor at unsteady state. Reduction of the model to dimensionless terms facilitates the testing of model sensitivity to important scaleup parameters and the analysis of various leaching rate regimes. Data from experimental column leaching of an artificial silver ore with cyanide solution, maintained under cyanide ion concentrations low enough to be rate controlling, are compared to the model, and their relationship is discussed.

METALLURGICAL TRANSACTIONS B VOLUME 24B, DECEMBER 1993-- 1087

II. M A T H E M A T I C A L F O R M U L A T I O N

The heap leaching operation is a heterogeneous, non- catalytic, fixed-bed reactor operating under unsaturated flow conditions. Lixiviant solution enters at the top of the heap and trickles through the interstices of the ore particles. Reagents diffuse into the pores and fissures of the ore particles and are gradually consumed by reaction with one or more solid reactants. These solids, in turn, are gradually dissolved throughout the heap. Based on this physical picture, shown schematically in Figure 1, a global heap model is derived with the following assumptions.

(a) The heap leach reactor is essentially an unsteady- state plug flow reactor, i .e. , flow of reagents and dissolved products occurs only by axial convection with no short-circuiting. (b) All physical parameters within the heap remain uniform and constant throughout the leach cycle. (c) No inhibition of diffusion into or out of the particles occurs due to dense packing or a stagnant boundary layer. (d) The heap operates isothermally.

At the particular level, the following assumptions are made

(a) The particles are spherical and of uniform size, density, and porosity. (b) All effective diffusivities are constant and uniform. (c) Reactive solids are evenly distributed throughout the pore surfaces of the particle, and their total relative volume is insignificant. (d) Dissolution reactions are irreversible, first order in the concentration of one rate-controlling reagent and variable order in the solid concentration of the reactant. (e) Intraparticular processes are not at steady state.

These assumptions at the particle scale result in the for- mulation of the general model which is the subject of a previous article, u21 In addition to the equations derived in that article, it is useful to account for the diffusion of dissolved products from ore particles and into the bulk

lixiviant of the heap. Assuming a one-to-one correspon- dence between moles of solid reactant i and its dissolved species, a mass balance for the dissolved species within the porous sphere is written

r ozci 20Ci] OCi O i e l L ~ F 2 + -~ ~ F | ~- " ~ - E~ kpi f~pf i fA = IsO - -

J Ot

where Ci is the concentration of dissolved species i, Die is the effective diffusivity of the solute within the particle pores, and all other terms are as previously defined, u2j This equation has initial and boundary conditions

Ci(r, O) = 0

Ci(R, t) = Ci~

OC'(o, t) = 0 Or

where Cib is the concentration of dissolved species i in the bulk lixiviant solution external to the ore particle.

Consider a packed bed of porous ore particles of radius R through which is passed a lixiviant solution at a constant flow rate. Assuming ideal plug flow through the bed and ideal mass transfer from the bulk solution into the pellet pores, the following mass balance may be written for dissolved species i in the bulk lixiviant which accounts for both the intraparticular and panicle surface reactions:

OC,b 3 ( 1 - - eh) [ e~. /OC,~ ] OC,b --tl s - - "Jr- - - ksiCsi CAb -- D i e [ - - I I = e ~ -

Oz R L \ Or/r=~J Ot

where us is the superficial velocity of lixiviant flow through the bed, e h is the void fraction of the bed (not including particle pores), and eb is the relative volume of the bed occupied by bulk lixiviant (generally a small number). This equation has initial and boundary conditions

Cib(z, O) = 0

Cib(O, t) = 0

C,.(Z,t) C,,(Z,t)

METAL DEPOSITS:

#o PORE / /

F i g . 1 - - S c h e m a t i c d i a g r a m o f a h e a p .

ORE PARTICLES V = (1 - ~) .Vb

ENTRAINED AIR V = (~h - E0.V~

BULK LIXIVIANT V = ~ - V k

ORE MATRIX v = (1 - ~0-(1 - ~.).vh

PORE STRUCTURE V = (1 - ~b)-~~ b

1 0 8 8 - - V O L U M E 2 4 B , D E C E M B E R 1 9 9 3 M E T A L L U R G I C A L T R A N S A C T I O N S B

An analogous mass balance equation for reagent A in the bulk lixiviant may be written when reagent A is strictly a component of the aqueous phase

OCAb 3(1 -- eh)

Oz R

aC Ab -- 6 b - -

Ot

lls ksiCsi CAb Av Dae

bi r=R

with initial and boundary conditions

Cab(Z, O) = 0

Cab(0, t) = CAO

A . The M o d e l in D i m e n s i o n l e s s F o r m

The preceding equations are recast in dimensionless form in order to determine the important design parameters. In addition to the dimensionless variables defined in a previous article u21 for the particle-scale leaching model, the following are defined for the global heap model:

Ci Cib z Ust Xi -- - - Xib -- ( = - 0 -

b i f Ao biCAo Z ebZ

New dimensionless parameters for the global heap model include

Die e b 3(1 - - 8h)DAeZ ~i -- V= 03=

D Ae eo(1 -- eh) u , R 2

where 6i is a diffusivity ratio of dissolved species i to reagent A, v is the volume ratio of bulk lixiviant to solution held up in particle pores, and w is the ratio of the porous diffusion rate of reagent A to the axial convection rate of lixiviant in the heap and corresponds to the inverse of the Peclet dimensionless group for mass transfer.

In addition to the equations which describe the particular leaching model in dimensionless t e rms , [121 the mass balance for dissolved species i within the pores of a particle is written

0 2 + ] - |_--7---~r xi 2 ox, 31- KpiO.~piPiol ~_ 0/~, _ 3 0/~, ~iL Or r oe J a.c v ~ ao


x,(~, 0) = 0

X/(1, r) = Xib

OXi ---7(0, "0 = 0 o~

In all of the particular model equations, the dimensionless diffusion time r must be replaced with the dimensionless bulk flow time 0 so that the particular and global heap models may be solved with a common time variable.

In the bulk lixiviant of the heap, the mass balance for dissolved species i is written

O'~ib I t ~ t Ksi Osi l OXib OC (1) ~i 30SiOtbJ = O0


X,b(L O) = 0

Xib(O, O) = 0

The analogous balance equation for a strictly aqueous- phase reagent A in the bulk lixiviant is written

,o + Z = off k \ a~5/e=, i~, o0


ab(~', O) = 0

,~b(O, O) = I

B. I m p o r t a n t M o d e l Func t iona l s

While the dimensionless model equations involve the fewest number of parameters, the solutions to these equations, in concentration vs heap depth and time, are of little use for designing heaps. The functionals which will prove the most useful include fractional conversion (integrated here over the entire depth of the heap), extraction, and heap effectiveness factor.

Once again, fractional conversion represents the fraction of ex trac table solid reactant which has been dissolved at a given time and is simply the particle conversion equation u21 integrated over the total depth of the heap. For solid reactant i, this is expressed

, ]

As in the particle model, X,. takes values from zero to o n e .

Extraction of reactant i from the heap represents the total fraction of extractable reactant i which has issued from the heap at any given time. Except for very short leaching times or in heaps with an unusually high holdup, extraction takes approximately the same value as conversion as a function of time but may be written simply as a function of the bulk concentration of dissolved species i at the bottom of the heap; thus,

F 0

Ei = ~iP Jo Xib( l ' O) dO

Unlike conversion, extraction may be evaluated directly from chemical analysis of heap effluent and is therefore a far more convenient measure of heap performance. It must be emphasized that these functionals are based on only that amount of reactant which can be obtained from the heap and not the total grade of the ore.

The heap effectiveness factor is defined after Ishida and Wen, u3J analogous to the particle effectiveness factor, t~2~ as the reaction rate of the entire heap taken as a fraction of the rate which would be obtained if there

M E T A L L U R G I C A L T R A N S A C T I O N S B V O L U M E 24B, D E C E M B E R 1 9 9 3 - - 1089

were (1) no diffusion limitation into the ore particles and (2) no convection limitation through the heap; i .e. , if the heap were a stirred tank reactor with reagent concentration maintained at Ca0. As such, the heap effectiveness factor may be written by integrating separately both the numerator and the denominator of the particle effectiveness factor equation, t~2/For solid reactant i, the resulting expression takes the form

3Kpi o'~Sa~2d~ 3i- KsiOrsiS*OIb d~

71' : I 1

3Kei '~2dsr + K,~cr, i dff

and, like the particle effectiveness factor, assumes values between zero and one.

III. N U M E R I C A L M E T H O D S

Details of the solution of all particle-scale model equations may be found in a previous article. 1121 All global heap model equations, including concentration gradients of reagent and dissolved complexes in the heap bulk solution, are transformed into ordinary differential equations through the definition of the substantial rate derivative, t~4J expressed in dimensionless coordinates

D 0 0 - +

DO 30 Off

This transformation facilitates the mathematical treatment of the global heap model from the Laplacian view- point, i.e., via the tracking of fluid elements through the heap. For this, simple backward finite difference ap- proximations are used: ~lSJ

Du 1

DO 6~ (btk,J - - Rk 14)

KU : K ldk_ l , j

where k is the heap depth index and j is the time index. All integrals are evaluated by compound quadrature

formulasJ 161 Integrals over space variables, including fractional conversion and heap effectiveness, are evaluated by Simpson's rule:

f ( x ) dx ~ s~ = ~ [f(xo) + 4f(x~) + 2f(x2) + 4f(x~)

+ . . . + 2f(xM-2) + 4f(x~_,) + f(XM)l

where

b - a h - , x ~ = a + j h , ( O < - j < - M )

M

and M is any even integer. Extraction, a time integral, is solved by the Trapezoid rule:

b h f ( x ) dx ~- tu = ~ If(x0) + 2f(x0 + 2f(x2)

where

+ . . . + ef(xN_,) + f(xN)]

b - a h - , x j = a + j h , ( O ~ j < ~ N )

N

and N is any integer. The order of solution of the model equations is as fol-

lows: (1) all initial conditions are defined; (2) starting at time 0 = 0 and heap depth ff = 0, the particle surface reactant concentrations o-si are determined; (3) the intraparticular concentration profile of reagent A, a vs ~, is determined given a value of the time step 60, and these values are used to solve for the concentration profile of the dissolved complexes, xi vs so; and (4) the intraparticular reactant concentrations r are updated. After the particular model equations have been solved, the particle surface concentration gradients are calculated, and these and the surface reactant concentrations are used to solve the global heap model equations, given a depth increment 6~, for the bulk solution concentrations of reagent A, ab, and the dissolved complexes, Xib, at ~" +

This procedure is repeated until ( = 1; then the model functionals are calculated, and the process is repeated for the new time, 0 + ~50.

IV. E X P E R I M E N T A L

Figure 2 shows a schematic diagram of the experimental leaching columns. The tall column was in six 61-cm (2-ft) sections made of transparent acrylic pipe of 9.53-cm (3.75-in.) ID, and the short column comprised only one section 30-cm (12-in.) long with 6.35-cm (2.5-in.) ID. All column sections were fitted with plex- iglas perforated plates and funnels with hoses at the bottom to collect and distribute lixiviant. Approximately 2 cm of glass wool were packed into the bottom of each column section to prevent the movement of any fine solids from one section to another. All column sections were open at the top.

Initially, the column sections were flushed with water to wash out any fine material which may have resulted from packing the ore pellets and to fill the pellet pores with water. Then, water was pumped into the column, and the flow rate was adjusted to 3.8 m L / m i n in the tall column and 4.1 m L / m i n in the short column corre- sponding to superficial velocities of 8.9 • 10 -4 and 2.2 • 10 -3 c m / s , respectively. Once the flow was steady, lixiviant was introduced. The lixiviant was a solution of 10 -3 M sodium cyanide, made basic with 10 -3 M sodium hydroxide to prevent the formation of hydro- gen cyanide gas. Concentration of cyanide was low enough to ensure kinetic rate control by cyanide ion concentration at all times. 117~ Solution samples were taken daily from the bottom of all column sections and analyzed for dissolved silver concentration using atomic ab- sorption spectrometry.

The ore consisted of artificial pellets containing 95 wt

1 0 9 0 - - V O L U M E 24B, DECEMBER 1993 METALLURGICAL TRANSACTIONS B

= Peristaltic pump

r

E--L___

Cyanide solution

i

,)

I i I

- - o r e

_ _ g l a s s w o o l

- - f u n n e l

waste waste

Short column Fig. 2--Schernatic diagram of the experimental apparatus.

Tall column

pct assay sand, 2 /3 of which was left coarse and 1/3 which was pulverized fine (to decrease the porosity and increase the adhesion of the agglomerates); 5 wt pct Type I1 Portland cement, plus pure silver metal powder, ranging from 4 to 7 /xm in diameter. This mixture was agglomerated with tap water on a disk pelletizer, pro- ducing spherical pellets ranging from about 0.5 to 2 cm in diameter. These were then cured under cover at room temperature for several weeks and became quite hard.

Porosity and density of the ore pellets were determined as follows. Several pellets were dried, smoothed by rubbing, measured with calipers, and weighed. They were then set in beads of water which was readily absorbed by capillary action. After a few minutes, they were dabbed to remove any excess water and weighed again. The porosity of each pellet was then determined by calculating the volume of water which was absorbed and dividing by the calculated volume of the pellet. Ore density was determined by dividing the dry weight of the pellets by their calculated volume and by one minus the calculated porosity. From these tests, G = 0.20 + 0.02 and Po = 2.1 --- 0.1 g / c m 3.

Column void fraction and bulk solution volume fraction were determined as follows. An empty column section was tared, filled with dried ore pellets, and weighed. The bulk density was calculated, and the column void fraction was determined by subtracting the ratio of bulk to pellet density from one. The column was then wetted, left under a steady drip for 1 hour, and weighed again. The amount of water in the pores was

calculated from the ore porosity and the column void fraction, and this was subtracted from the weight of water in the column. The volume of the remaining bulk water was divided by the volume of the column to obtain the bulk solution volume fraction. From these tests, eh = 0.39 and G = 0.03.

V . R E S U L T S A N D D I S C U S S I O N

A. Computer Simulation: One Solid Reactant

In order to examine the general behavior of the model, the following restrictions were temporarily applied: (1) only one solid reactant (n = 1), (2) first-order rate dependence in the solid reactant concentration (~bp = 1), and (3) no surface deposits (3. = 0). The diffusion equation for reagent A now becomes

OPa 2 Oa 3 Oa - - + - - - - Kppa = - - - - O~ 2 ~ a~ I'~W aO

and the rate expression for the disappearance of solid reactant

d%_ Kd3,,o~ - - o',,c~

dO 3

The convection equation for reagent A through the heap is simplified to

Oab w(Oa~ _ = Oa___2

o~ \asU~=~ o0

METALLURGICAL TRANSACTIONS B VOLUME 24B, DECEMBER 1993--1091

and these three equations have initial and boundary conditions

o~(s ~, 0) = 0

a(1, 0) = c~

c~a - - ( 0 , 0) = 0 0~

%(~:, 0 ) = 1

a~(~', 0) = 0

O~b (0, 0) = 1

and require four parameters for their unique solution, Kp, /3, z,, and to.

Heap concentration profiles of reagent A (solid curves) and fractional conversion profiles of one solid reactant (dotted curves) are presented for all combinations of Kp = 1, 10, and 100 and the product Kpto = l, 10, and 100 at /3 and u = 1 in Figure 3. At Kpto = 1, which represents a short heap operating at a high lixiviant flow rate, reaction occurs at a fairly uniform rate throughout

the heap, regardless of the value of Kp. This is as expected since a low value of xpw suggests a fast convection rate relative to reaction and, thus, negligible convection resistance. Hence, at low values of Kpw, the global model of the heap is largely unnecessary, and only the particle model is needed to simulate the leaching process.

As to is increased at any value of Kp, the convection limitation becomes more significant. At Kpto = 10, the reagent concentration profiles are monotonic, and convection through the heap may or may not be the rate- limiting factor, depending on the actual value of to. At t%to = 100, the reagent profiles are sigmoidal, with only a narrow region of the heap under active leach at any given time, and the degree of reactant conversion at any depth parallels the propagation of reagent to that depth. At these parameter values, heap leaching is completely convection controlled, and one is left with what might be called "shrinking-heap" kinetics.

Figure 4 shows plots of the effluent complex concentration with 6 = 1 (solid curves), overall fractional conversion (large dashes), and extraction (small dashes) as functions of flow time for Kp = 1. At /3 = 0.01, the number of residence times required for complete reaction is large, and with so much fluid being passed

Xp = 1 Kp = 1 0 ~:p = 1 0 0

1.o &8 = 1 1.0 &O = 1 ~.o &8 = 5

o, o, ......... i! 0 . . . . . . . . . .

XpW = 1 ~ . -__ 0.4 0.4 . . . . -

o , . . . . o , o ,

o . o o . o ~ ~ oo o.o .7..r,.r,,,7.,=.r..,7,,=,r,,,_,7..r.,.7...-. o.o o.2 o.* o.~ o.'8 1.o

l o &8 = 0.1 '~o &O = 0 .5

o., ~ o.s

I 0 .6 0 .6

0.2 0.2

' 0 .~1 ' 11' 161~1 ' ' ' ' I ~ .~ r ' ' ' ' a }~ ' ' ' ' I l a ' g ' ' I I I ' i ' g 0 0 ~ ~ 0 ~ 0 ~ 0 ~ ~ 0 " 0 . 0 0 .a 0 .~ 016 " 110

Kp6.) =

A8 = 0.1 1.o

'i!i I .

l O O g

I - i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . ~ . . . . . . . . . f . . . . . . . . . i

o.o o .z o . , o .e o.8 1.o

Dimensionless depth, ~"

AO = 0.1

l l ~ l t I I I 0.6

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

Dimensionless depth, r

A0 = 0.1 1.0

i!i 0.0 0.2 0.4 0,6 0.8 1.0

Dimensionless depth, r

Fig. 3 - -Solu t ions to the simplified continuity equations for reagent A in the bulk solution (~b) and the fractional conversion (X) as functions of heap depth.

1092--VOLUME 24B, DECEMBER 1993 METALLURGICAL TRANSACTIONS B

1.0 4 1.0

laJ 0.8

0.6

X o . 4 I I 1 0,2

0.0

/ J

/ / /

/

/ / ;P =- 0.01 / o~ = I

1 v = 1

/ / , r . . . . . . . f . . . . . . . . . i . . . . . . . L , i . . . . . . . . . i . . . . . . . . . 201 401 601 801 1001

Dimensionless f low t ime, 8

b I O, 8 ///'"

/ /

/ /

/ ,' ,r = 1 / I ~P= 1 / /

/ ,, ~ = 1 / , ' u = 1

/ ,'

o.o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 9 13


%! ] \/ ~ : 4 /'~ xp= 1 % . o.6zl , / \ p = O.Ol

o.o ~ 1 ~ ; . . . . . . ~ . . . . . . . ~( D i m e n s i o n l e s s f low t ime, 8

L 1.0 t

, 0 . 8 1 ~ ! 0.6

,, ~

0.0

, - . - 5 : - r----"

/cp = 1 ~ / / , ' " fl = 1 ,," w = 10 / . - / X . ' "

/ .

f l H r r H i I , 1 . 0 1 . 5 2 . 0 2 . 5 3 , 0 3 . 5

D i m e n s i o n l e s s f low t ime, 0

1.0

, 0.8

, 0.6

X

;~11 0.4 1 I 0.2 =

0 . 0 1

/ , / / / /

/ / /

/ / / / / top = 1 0.01

/ // /2 =

//

41 81 Dimensionless f low t ime, 0

121

t J J / s '

, o.a P -- 11 t - " i / " / i I

~._J! 0.6

[ / / / t

I 0 . 2

/ / /

0 . 0 / / / ' 1.0 1.5 2.0 2.5 3.0


Fig. 4 - - E f f l u e n t concentration (Xb(1, 0)), fractional conversion (X), and extraction (E) vs dimensionless flow time (0) given various values of fl and w.

through the heap over the course of the leach cycle, the total holdup of dissolved products is negligible. Hence, the conversion and extraction curves largely coincide at all values of Kp and oJ. At KpCO = 1, the effluent concentration curve consists of a short peak followed by a long tail, and the conversion curves are roughly the shape of exponential response curves, signifying the dominant role of particle kinetics. As KpW is increased, the Xb peak grows taller until, at Kpto = 100, it reaches its maximum value of one and flattens out. Above a cer- tain value of to, the dimensionless flow time for complete reaction decreases to its limit, 0 = 1//3v, which is simply the time required in order to feed a stoichiometric amount of reagent to the heap. The conversion curve is a straight line, since the rate of propagation of the reaction zone through the heap is constant at d ~ x / d O ~- ~3to ( e . g . , the bottom left comer of Figure 3).

At /3 = 1, the overall flow times are much shorter, and a "chromatographic effect" due to the high diffusion driving force at the beginning of the leach cycle results in significant holdup. This effect which causes the

characteristic delay in attaining peak concentration in all heap leaching operations is only significant when the ore grade is very low, as in precious metal leaching operations. At KpW = 100 and Kp = 1, both the conversion and extraction curves are linear, yet nearly half of the solid reactant has been converted before any dissolved complex is seen in the heap effluent.

The relative magnitudes of convection, diffusion, and reaction resistance are most conveniently illustrated in semilog plots of heap effectiveness factor vs fractional conversion, as shown in Figure 5. At low values of to, the heap effectiveness factor approaches a limiting value dependent on the particle kinetics, u2] As to is increased, the effectiveness drops until, at approximately Kpto = 100, the ineffective character of the bulk flow process dominates, obscuring any effects at the particle level. This happens irrespective of the reagent concentration of the incoming fluid.

Relaxing the restriction of no surface deposits and also assuming first-order rate dependence at the surface, the surface rate expression becomes


o

0.1 :?

S -

4 -

2 -

0.01 0.0 0.5

0.1

0.01 1.0

2 -

~ - - 0 . 1

0 . 1 ,=-='s'7-"~Z

a -

= h--p = 1 O0

0.01 . . . . i , , , , 0 .0 0 .5 1.0 0 .0 0 .5 1.0

Frac t iona l convers ion , X

Fig. 5 - Heap effectiveness factor (-q) v s fractional conversion (X) for the continuity equation solutions shown in Figure 3.

do" s K~Bvto - - O ' s a b

dO 3h

and the intraparticular rate expression becomes

d o ' p __ K p f l V W - - O'pO~

dO 3(1 - A)

These equations have the following initial conditions:

~({:, 0 ) = l

o~,(~r, o) = 1

Since it has already been determined that surface deposits have little or no effect on the kinetics of particles with low Kp, t]2j this study only examines the case when this parameter is high and particle kinetics are diffusion controlled. Figure 6 shows the effect o f various surface fractions h on the rate o f reactant conversion from the same particles in two different heap simulations. Higher surface fractions result in much faster conversion rates at to = 0.1 but have little impact at to = 1. At low to, kinetics at the particle level control the entire process, so the leaching rate is strongly enhanced by higher surface fractions. At the higher to, however, the characteristic contours o f the particle kinetics all but disappear as the rate of bulk flow through the heap becomes rate controlling.

Similar results are obtained by varying the reaction order ~bp, as shown in Figure 7. For these plots, a low value o f Kp for the particles is assumed since reaction order is only a factor in systems under chemical reaction control, t~21 Again, the effect is greater at low values of to, especially at the beginning of the leach cycle. How- ever, the suppression of particle kinetics at the higher value o f to is less pronounced, since a higher reaction order tends to widen the active reaction zone in the heap, A~" x, resulting in a net increase o f heap effectiveness with reaction order.

B. Computer Simulation: Competing Solid Reactants

In a previous article, it was shown that the presence of one or more competing reactants within porous ore particles can significantly alter the kinetics o f conversion

X 1.0

~ 0 . 8

> ~ 0 . 6

0 0.4

--~0.2

L _ 0.0 b_

1.0

1.0

0"~. 4

0.2 0

//// u - 1

o.1 i i i i 1 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i J i i i i

2 3 4 5

D i m e n s i o n l e s s f low t ime , O

X

~ 0 . 8 0

~

C O L . -

0 0 . 6 > c- o o 0.4

1D t--- 0 0 . 2

, - -

0

0.0 I, 1.0

X = I

X = 0

'-11oo -11

i i i i I i i i i I i i i i I 0 i i i

1.5 2.0 2.5 3�9

Dimens ion less flow t ime, 0

Fig. 6--Fractional conversion (X) v s dimensionless flow time (0) given various surface fractions (h).

of a valuable mineral component , l~2] Computer simulation of the particle model showed how the presence o f a reactive gangue material may push the otherwise highly effective reaction o f a valuable component into a much less effective regime and how the overall process may approach a near-steady state. Figure 8 shows the concentration gradients o f reagent A and one solid reactant with Kp~ = 1,/31 = 1, and ~bp~ = 2 /3 with no surface deposits in a heap at w = 1 and v = 1. A value of ~bpt = 2 / 3 implies that solid reactant 1 is present as dis- creet spherical blebs. In one simulation, reactant 1 is alone, and in the other, it is accompanied by a reactive gangue material with Kp2 = 10, f12 = 0.01, and thp2 = 0, also with no surface deposits. At these parameter values, identical to those chosen in the previous article, it is assumed that there is 100 times more solid reactant 2 than 1 but that its intrinsic dissolution rate is 10 times slower. Also, a value o f thp2 = 0 implies that the gangue material forms an even coating over the internal surfaces of the particle, or perhaps it is the ore matrix material�9

Alone, reactant 1 dissolves effectively throughout the heap, and the reagent gradients are insignificant. In the presence of solid reactant 2, however, the reaction shifts more toward the top o f the heap, and the reagent gradients are steep, with ab < 0.12 at the bottom�9 In addition, the reagent gradients within the heap approach a


t . 0 X

~ 0.8

~ 0.6

tO 0.4

-~00.2

L 0.0 14_ 2 3 4 5 6 7 8 9 10

Dimensionless flow time, e 1.0

X

~ 0 . 8

. ~ 0 . 6

g 0 0.4

0 0.2

(] L 0.0

t J_

/ ~ p = 2.0

/ / 2 3 4

D i m e n s i o n l e s s f l o w t i m e , (3

Fig. 7- -Frac t iona l conversion (X) v s dimensionless flow time (0) given various orders of reaction (~bp).

steady state, just as they do within the pores of the individual particles. Hence, the conversion kinetics of reactant 1 could be approximately described in this case by a steady-state solution for reagent A, using only the parameters for reactant 2. As outlined in the previous article, the diffusion equation for reagent A becomes the spherical Bessel equation

O2ct 2 da d~ 2 + ~ dE Kp20g 0

coupled with the mass-action rate expression for reactant 1

dOrpl Kpl/~ 1 b'to

dO 3 - - r ~

~pl

Additionally, the heap convection equation for reagent A becomes

OOfb /O( 00~ - - + = 0

An analytical expression for the concentration of reagent

A as a function of particle radius and heap depth is now easily obtained and takes the form

sinh (~f~Kp2~) .,-ir e s c ~ 0

~: sinh (~p~p2)

e ~ = 0 sinh ( 'V~p2)

where

X~p2 - tanh (V~.2) y =

tanh (X~p2)

The concentration of solid reactant 1 is expressed

c, f f , (, 0) = 3

exp ( - KP~C33'v~ ) ~(~, ~0 ~, =

For our test situation, these equations become

sinh ( X / ~ ) --O~(ff, ~ 5 ~ 0) = e -217~

11.8~:

a(~', ~ = O) = 0.268e -217~

Orb( ( = 1) = 0.114

o-p,(~, ~, 0) = 1 -

The conversion curves for reactant 1 both within and without the presence of reactant 2 are shown in Figure 9. As expected, the conversion when n = 2 is significantly delayed relative to when n = 1. The effect of the second reactant on the heap effectiveness factor of the first reactant r h is illustrated in Figure 10. When n = 2, the r h - X1 curve does not approach unity near complete conversion, identical to the particle model alone, uzl However, in a very ineffective (high to) heap, this would not necessarily be the case, since the second reactant would have to be depleted at every depth before the reaction zone could move downward into fresh ore. Under these extreme circumstances, the time for total dissolution of reactant 1 (or any other minor constituent) would be 0 ~ 1//32v.

C. Computer Simulation: an Approach for Particle Size Distributions

In a previous article, uzj the particular leaching model was solved for a distribution of particle sizes based on the Gates-Gaudin-Schuhmann distribution function. In this study, these results are applied to the global heap model, albeit limiting our discussion to ore with no significant surface deposits.

If a dimensionless particle radius is defined as

R _ , ~ = - -

METALLURGICAL TRANSACTIONS B VOLUME 24B, DECEMBER 1993--1095

1.0 x 1.o

X g o . 8 n = 1

I 0 . 8 "~ n = 2

I ~ 0 . 6 I i 0.6 o o

0 .4

~ / / ~; = 1 ~2 = o.o l o , o = = o

0 .2 L~_ o.o 1 2 1 41 6 1

D i m e n s i o n l e s s f l o w t i m e , 0

O.O Fig. 9 - - F r a c t i o n a l conversion of solid reactant 1 (XO v s dimension- 0 . 0 0 . 2 0.4 0 . 6 0 . 8 1 . 0 less flow time (0) for the sytems shown in Figure 11.

Dimensionless depth, {" 1.0

7 \~ "- "- - -, "-> "-'~ ~- ~--~ X "1" .k\ \" "- ~ - -, " , " " - ~ " ".q'-~"~-

1 ~,kk x \ x \ \ \ ' - \ \ , ~ k . ~ . I 0 . 8 - t , \ ~ \ ", \ \ \ " . \ \ , ' , , , \ ~ . , - I " . X X X " " , , \ ,, , , ' , \ \ - .,,,,,,x ~.. I 7 " . ' k ~ \ \ \ \ \ \ " 2. \ " .'- \ \ '~ ~.

l " \ ~ " " ", " ,, ",." "- ' -" ",.,.,.": I 3" ~ . ~ \ \ \ \ ", . , . \ \ \ ' , \ ~ \ . , 0 .6 j \ \ - , ~ ,, \ \ \ \ \ \ \ \ ,, -I \ \ % . ~ \ \ \ \ \ \ \ . x \ \ 4 " N ' % . \ . \ " , \ \ \ ' \ 4 "" \ ' ~ \ \ \ \ - . \ \

g 0 . 4 J \ \ ~ . \ \ \ \ \ x J'~ \ \ ~ \ \ . , \ \ J ~ - . \ ~ . - , ~ ~

0 . 2 ~ ~

0 . 0 - - I ' | I | | i I i ' , I i , i i I i i i i I i , i ,

0 . 0 0 . 2 0 . 4 0 .6 0 . 8 1 .0

Dimensionless depth, Fig. 8 - - S o l u t i o n s to th, continuity equations for reagent A (~b) and fractional conversion of reactant 1 (X0, alone (n = 1) and in the presence of a second reactant (n = 2). Kpl = 1, ~1 = l , ~pl = 2/3 , Kp2 = 10, /32 = 0.01, ~p2 = 0, ~o = 1, V = 1, and A0 = 1.

where/~ is some reference radius, then for any normal- ized weight fraction distribution of particles w(.~)

~mi[ X W ( ~ ) d E = 1

where E.,~, and ~=m~• are the minimum and maximum dimensionless particle radii, respectively. Assuming the simplest case of no mineral liberation upon comminu- tion, as in the previous article, and also ignoring surface deposits, the rate constant Kei, the diffusion time r, and the global heap parameter ~o will vary as functions of particle size; thus,

Kp~-- ~2gpi r =2_ oJ =2

C o 0 0

tO 01 (D

C

>

0 (D

4 . - (D

13 (D

-1-

0.1

8 -

7 -

8 -

5 -

4 -

n = 1

2-It \ n = 2

; pll = 11

~opl = 2 / 3

= 101 99p2 = 0

I ' } w = 1 ~ = 1 " , , , , , , , , , i , , , , , , , , , 0 .0 0 .5 1.0

Fract ional conversion, Xl Fig. 1 0 - - H e a p effectiveness factor (Th) v s fractional conversion of solid reactant 1 (Xt) for the systems shown in Figure 11.


where Kpi, Z, and o3 apply to the reference particles. Sub- stitution of these altered parameters into the particular leaching equations allows one to solve the particle model for a range of particle sizes E based on the parameters from the reference size E = 1. In order to gage the effect that a distribution of particle sizes will have on the heap leaching process as a whole, the convection equations must be altered. Employing the GGS distribution, uS/ these take the form

OXib OXib lZmax ( _ _ + _ _ _ o3 a x , w(E)

a0 O~ r J-=m,, \0~:/~=, E - - - T d E

= -o3 \ a~:]r m E m-3 dE

--+ - ~ J-zm t-0-~) 30 3( ~. ~=t E2 dE

f(0o) = - w Jo \ a U e = , mE~-3 dE

where m is a parameter of the GGS distribution and the initial and boundary conditions are the same as before. Likewise, the fractional conversion becomes

L Lfo X,---- 3 m ~ , m l(1 - ~e,)~2dMEd~

and the heap effectiveness factor, in the absence of surface deposits, is written as

m_ o'pf'a~ d~d-dff JO dO dO

~i ~ 1 1 1 LLfom~m+'o'~/"2d'd~d' Figure 11 shows plots of effluent dissolved species concentration vs dimensionless flow time calculated for both a particle size distribution (m = 1, dashed curves) and the reference size alone (m = ~, solid curves). As would be expected, the difference between the two situations is larger at Kp = 100 than at Kp = 10, since particle size has more bearing on the reaction rate at larger Kp values. The effect is obscured at high ca values, since the effluent concentration is already at or near its upper limit. At low ca values, the xb(l, 0) curves are simply a little higher in the size distributed system than in the isometric one. The most drastic difference is at moderate ca values, where the entire shape of the effluent response may be dramatically altered.

D. Experimental Column Leaching Tests

Table I summarizes all of the parameters for five column leaching tests, the results of which are shown in Figures 12 through 16. All of these parameters were either measured directly or calculated from known values, unless otherwise noted. Values of Da~, kp, and &p were determined in separate batch kinetics tests. The surface fraction h was taken as zero. The value of the

~ 1 . 0 ;f -% - m = oo

~-- " " . , m = 1 " -s163 ~ 1 / ' - - - ~ ' , , - = 1 0 ? X I ~__ 1 ~ ~ ~ = lo = 0.1 v = 1 { jO.6

o 0 0 . 4

02 ;1 o , ",

"~o.o :,%, " " 7 " ~ -~'-- i i i i i i i i i i i i i i i i i i i i I l i i i i i i i i i I I I I I I l I T I

r~,1.0

6 11 16 21 26 Dimensionless flow time, 0

t / f ' - - - ~ - \ . m = oo ~1! \,, m=1

,~ \ '? = ~ ~0 = 1oo " ~ 0 . 8 ~ ] - , \ l fl = 0.1

~ 0 . 2 ~. ~ = 0.01 " ~ . _ -~_

"oo ;_ii 55-;; 1 11 21 31

Dimensionless flow time, 0

Fig. l 1- -Eff luent concentration (Xb(l, 0)) vs dimensionless flow time (0) given various values of o3, for a single particel size (m = ~) and a GGS distribution (m = 1).

GGS distribution parameter m is based on a random sample of 100 pellets from column test 2 and assumed valid for the other tests. In all of the tests, o3 is treated as an empirical parameter chosen on the basis of the best fit of the experimental data. This value is denoted o3app, and the value predicted from theory for comparison is denoted o3p,.

The "expected" values of CAO and C~ resulted from adding known amounts of sodium cyanide and silver powder to the lixiviant solution and the ore pellets, respectively, while the "observed" values were derived from the effluent responses of the individual column tests. Hence, for the model fits, 100 pct extraction was based on the observed values of Ceo in order that the extraction integrals of the data and the simulations be equal. "Actual" extractions ranged from 80.9 pct in the tall column (test 1) to 98.3 pct in the low-grade pellet test (test 5) and were a function of pellet grade and particle size. The parameter/3 was calculated based on the observed values, while gp was based on the expected values.

Effluent profiles from the tall column test (test 1) are shown in Figure 12. This column simulates an ineffective heap (Figure 17), since the effluent response is at the maximum value of Xb(1, 0) = 1 over most of the


Table I. Parameter Values for the C o lumn Leach ing Tests (All Units in the cgs System) b = 0.5 eh = 0 .39 DA, = 2.35" 10 -6 eo = 0 .20 ko = 3 . 7 6 - 1 0 7 v = 0 .246 m = 6 p o = 2 . 1

= 1 (assumed) ~bp = 2 ~:b = 0.03

Column Column Column Column Column Parameters Test 1 Test 2 Test 3 Test 4 Test 5

Ca 0 " 106 (exp) 1 1 1 1 1 Cao" 106 (obs) 0.91 1 1 1 1 Cp0" I0 6 (exp) 6.34 6.34 6.34 6.34 0.634 Cp0" l06 (obs) 5.13 5.60 5.26 6.05 0.623 NRe 0.117 0.288 0.358 0.218 0.288 /q 0.655 0.655 0.814 0.496 0.655 u~.- 103 0.89 2.2 2.2 2.2 2.2 Z 366 30 30 30 30 flt*i 0.0106 0.0106 0.0113 0.0098 0.0955 /~p{**l 930 930 1430 530 9.3 a3~pp 1.1 O. 10 0.071 O. 13 O. 10 a3p, 4.12 O. 137 0.089 0.238 O. 137

*Based on observed values. **Based on expected values.

"' �9 ( = 1 / 6 o o o o o 1 / ,3 ,mii. 1 / .2 u u u u u 2 / 3 C o l u m n t e s t 1

i . . * * 5/6 A A A A A 1

Mode l f i t : #p = 9 5 0 , /3 = 0 . 0 1 0 6 rpp = 2, v = 0 . 2 4 6 ~ = 1.1, m = 6

1.o �9 ~ . . . . . Iz i" ""ig~--A"

0.8 i []

0.6 i

d~

o.4 , X~ _\ ' -

0 ,0 I I I I I I I I i I I I I I I I I 1 1 1 I

1 O0 200 ,300 400

Dimensionless flow t ime, 0 Fig. 12 - -Ef f luen t concentration and extraction data from column leaching test 1.

leach cycle. The apparent value o f o3 is 1.1, while the predicted value is 4.12. This large disparity is most likely due to a combination o f nonideal flow patterns (i.e., rivulet flow and stagnant zones) within the column and to the blinding of pellet surfaces due to close packing. The former would have the effect of increasing the local bulk fluid velocity in some regions, leaving others accessible only by dispersion along pellet interstices.

ILlo0 1.0

0 0 .8 0 0 . . . . . . . . .

06 \ / = 93o ~" ~ / ~ o = 2

V F= o.olo6 o.4 _/k, ~ = o.1oo

�9 0 . 2

�9 f �9 0.0 . . . . . . . . . . . . . . . . . . . . . . . ; 0 1 I'061 1501i . . . . . . . =~;'~

Dimensionless flow time, e

Fig. 13 - -E f f l uen t and extraction data from column leaching test 2.

IJl 1.0

o O o 0.8 o \ j 0 . . . . . . . . .

0.6 \ / E~ = 14~o

% / # = o.o115 ~ , 0.4 V ~ -- 0.071

><

�9 0.2

�9 0.0 . . . . . . , , i . . . . . . . . . i . . . . , , , , ,1510 . . . . . . . . 501 1001 1 2001

Dimensionless flow time, e Fig. ] 4 - - E f f l u e n t concentration and extraction data from column leaching test 3.

1098-- VOLUME 24B, DECEMBER 1993 METALLURGICAL TRANSACTIONS B

I L l 1.0

0 0 0 0 .8

0 0

0 .6

(E)

,,~. 0.4

~ 0 , 2

�9 0 . 0

~, p z, = 53o I - \ . / #p = 2

, 8 = 00_o98 / \ ~ = o.15o

501 1001 1501 2061 Dimens ion less f low t ime , 0

Fig. 15--Eff luent and extraction data from column leaching test 4.

0.20 1.0 O O

.0.8 (1) 0.15 'o 0

"~ \ o / - ~0 = 9.3 .0.0 o o ;~<0.10 \ / ~p = 2 ___ 0 �9 ~k/_ /9"= 0 0 9 5 5 o �9 . ~ X " ~ = 0.100 0 . 4 �9 r T]

�9 0.05 -0.2

0 , 0 0 . . . . . . . . t . . . . . . . . . , . . . . . . . . . ~ . . . . . . . . . ~ . . . . . . . . . 0.0 201 401 601 801 1001

Dimensionless flow time, 0

Fig. 16--Eff luent concentration and extraction data from column leaching test 5.

The latter would cause a decrease in the effective particle size. Both would account for the observed defi- ciency in o3, resulting in apparently higher column effectiveness while obscuring the effects of kinetics at the particle level.

Figures 13 through 15 show effluent response and extraction curves for three short column tests (tests 2 through 4). Each involve the same grade of ore pellets at test 1 but at three different particle sizes. As shown in Figure 17, these columns simulate fairly effective heaps based on the shape of the effluent response curves. Again, however, the apparent values of o3 fall short of the predicted values but not by as large a margin as in the tall column test. Finally, the results of a short column test with low-grade pellets (test 5) are shown in Figure 1 6. At the same 03 value as for test 2, the model fits the data quite well, even though the kinetics within the pellets of test 5 are mostly reaction rate controlled while those of test 2 are diffusion controlled. The sim- ulated extraction curve ends up significantly lower than the data, but this is probably as much the result of experimental error as of any physical phenomenon.

Overall, the results of these column leaching tests at- test to the validity of the plug-flow modeling approach. However, in order to predict the performance of actual

I,,,- o 0 0

CO 0 C 0 >

0

4--

0

0- 0 0

r"

I o -

6

5

4

3 -

2 -

f

0.1 B -

7 -

6 -

5 -

4 -

3 -

2 -

( T e s t 4)

(Test 2)

(Test 3)

0.01

7 -

6 -

sJ , 1)

3 -

2 J I i ! I ; I I " ] I I I i I ! ! I I I

0 . 0 0 . 5 1 . 0 F r a c t i o n o l c o n v e r s i o n , X

Fig. 1 7 - - H e a p effectiveness factor (rl) vs fractional conversion (X) for the model simulations of the experimental column leaching tests.

heap leaching operations from column tests, some sort of correlation between the theoretical p l u g - f l o w mode/ and the actual flow patterns within the heap is necessary. Assuming this correlation to be a function of such factors as bulk flow rate, particle size, reagent diffusivity, surface tension, e t c . , it may take the general form

o3app = f(o3pr, NRe, Nsc, Nwe, m . . . . )

where NRe, Nsc, and Nwe are Reynolds, Schmidt, and Weber numbers, respectively, m is a parameter of the size distribution, e t c . There is the opportunity, in the present study, of analyzing only the effect of the Reynolds number, since the same ore type and reagent species were used in all of the tests. Satterfield, Hgl in a review of the literature on the performance of catalytic trickle-bed reactors, defines the "contact effectiveness" as the ratio of the apparent to the predicted mass transfer coefficient and suggests a relationship, derived by Bondi, t2~ of the form

k~P----ZP = 1 - a L -b

kpr


1.0

0.8

I# 0.6

g_

o4

0.2

0.0

0.1

NRe Fig. 1 8 - -Co n t ac t effectiveness (aT, pp/aT~) vs Reynolds number (NRc) from the experimental column leaching tests.

where the k's are mass transfer coefficients, L is the mass flow rate, and a and b are constants. Modifying this general form to accommodate our parameters and defining the Reynolds number as

2Rusp N

R e ~ " - -

tx

where p and /x are the solution density and viscosity, respectively, our experimental results are found to ap- proximate the correlation

O~app = 1 - 0.066 NRe 115

O~pr

This result is shown graphically in Figure 18. Ob- viously, a value of one on the ordinate represents a situation in which all of the assumptions which went into the derivation of the global heap model are valid. Cor- respondingly, a heap with a serious short-circuiting problem would approach an ordinate value of zero in Figure 18. Thus, it makes sense for the curve to approach an ordinate value of one asymptotically with increasing Reynolds number, since faster flow rates (us) coupled with smaller specific particle area (3//~) would facilitate a more even distribution of lixiviant over particle surfaces. While certainly not enough data have been analyzed here to develop an empirical correlation with a wide range of applicability, at least the method of approach suggests an excellent avenue for future study.

VI. C O N C L U S I O N S

The proposed model is capable of simulating the heap leaching of one or more solid reactants from porous ore

particles. It is shown that the heap undergoes homo- geneous, high effectiveness kinetics for w ~ 1, zone- wise "shrinking-heap" kinetics for 6o >> 1, and one or the other when o~ ~ 1, depending on the value of Kp.

Factors which affect kinetics at the particle level, such as deposits of solid reactants on the external ore surfaces and the variable order of reaction, are shown to have a significant effect on the rate of heap leaching only in heaps with low 6o values. All kinetic resistances at the particle level are overshadowed by the bulk convection resistance in high 6o heaps.

It is also shown that a second solid reactant competing for reagent can decrease the reaction rate of the first reactant, and if a zero-order reaction may be assumed for a major reactant (i.e., a gangue material), then an analytical, steady-state solution to the model equations for any minor reactant is straightforward.

The model is also capable of simulating the heap leaching of particles with a distribution of sizes. Particle size distribution has the greatest effect in heaps with low to moderate 6o values and only when particle-level kinetics are diffusion controlled.

Results of experimental column leaching tests at a range of parameter values confirm the appropriateness of the plug-flow approach to the global heap model, even though the underlying assumptions might not be completely valid. The model provides an excellent fit to the experimental data if ~o is treated as an empirical parameter. It is shown that the apparent and predicted values of 6o may be correlated with Reynolds number, indicating that the deviations from ideality are largely a matter of fluid mechanics.

bi CA C~b CEiO

Ci

Cib

c~,0

Csi

Csio

OAe

Die

Ei

N O M E N C L A T U R E

stoichiometry number, mol i /mol A concentration of reagent A, mol A/cm} bulk concentration of reagent A, mol A/cm~ initial extractable grade of solid reactant i, mol i/g ore concentration of dissolved species i, mol i/cm} bulk concentration of dissolved species i, mol i/cm} grade of solid reactant i within particle, tool i/g ore initial grade of solid reactant i within particle, mol i/g ore grade of solid reactant i on particle surface, mol i/g ore initial grade of solid reactant i on particle surface, mol i/g ore effective pore diffusivity of reagent A, cm}/cm~ s effective pore diffusivity of dissolved species i, cm~/cm~ s extraction of dissolved species i, dimensionless rate constant of solid reactant i within particle, mol i/g ore s [Cp] e~ [CA]


ks,

m

n

N~e F

R t

Us

w

Xi

z

Z

rate constant of solid reactant i on particle surface, mol i/cm2p s [Cs] 6 [CA] Gates-Gaudin-Schuhmann size distribution parameter, dimensionless number of solid reactants pellet flow Reynolds number, dimensionless radius, cm particle radius, cm time, s superficial bulk flow velocity, cm}/cm 2 s Gates-Gaudin-Schuhmann size distribution function, dimensionless fractional conversion of solid reactant i, dimensionless depth, cm heap depth, cm

Greek letters

a dimensionless concentration of reagent A

ab dimensionless concentration of reagent A external to particle

fli reagent strength parameter relative to solid reactant i, dimensionless

"y modulus in steady-state model

6,- ratio of diffusivity of dissolved species i to reagent A, dimensionless

eb bulk solution volume fraction

eh heap void fraction eo ore porosity ~" dimensionless depth ~7i effectiveness factor for

solid reactant i, dimensionless

0 dimensionless flow time Kp~ Damk6hler II number for

solid reactant i within particle, dimensionless

Ks~ Darnk6hter II number for solid reactant i on particle surface, dimensionless

A~ surface fraction of solid reactant i, dimensionless

v ratio of volume of bulk fluid to fluid in particle pores, dimensionless dimensionless radius dimensionless particle radius ore density, g / c m 3 dimensionless grade of solid reactant i within particle

o - s i dimensionless grade of solid reactant i on particle surface

Po ~pi

~- dimensionless diffusion time

the i reaction order for solid reactant i within particle, dimensionless

t h s i reaction order for solid reactant i on particle surface, dimensionless

Xi dimensionless concentration of dissolved species i

X~ dimensionless bulk concentration of dissolved species i

o~ inverse modified Peclet number, dimensionless

Subscripts

A reagent A b bulk solution external to

particles i reactant species i h heap o ore matrix p within particle s on particle surface X at constant X

ACKNOWLEDGMENT

The authors acknowledge the financial support of the United States Bureau of Mines under the Mining and Mineral Resources Institute Program (G 1114132).

REFERENCES

1. R.J. Roman, B.R. Benner, and G.W. Becker: Trans. AIME, 1974, vol. 256, pp. 247-52.

2. D.G. Chae and M.E. Wadsworth: In Situ Recovery of Minerals, K.R. Coyne and J.B. Hiskey, eds., Engineering Foundation, New York, NY, 1989, pp. 1-21.

3. J.A. Harris: Proc. Australas. Inst. Min. Metall., 1969, No. 230, pp. 81-92.

4. L.M. Cathles and J.A. Apps: Metall. Trans. B, 1975, vol. 6B, pp. 617-24.

5. L.M. Cathles and W.J. Schlitt: Leaching and Recovering Copper from As-Mined Materials, SME-AIME, New York, NY, 1980, pp. 9-27.

6. H.W. Gao, H.Y. Sohn, and M.E. Wadsworth: Metall. Trans. B, 1983, vol. 14B, pp. 541-51.

7. H.W. Gao, H.Y. Sohn, and M.E. Wadsworth: Metall. Trans. B, 1983, vol. 14B, pp. 553-58.

8. R.L. Braun, A.E. Lewis, and M.E. Wadsworth: Metall. Trans., 1974, vol. 5, pp. 1717-26.

9. R.W. Bartlett: Int. Syrup. on Hydrornetallurgy, D.J.I. Evans and R. Shoemaker, eds., A1ME, New York, NY, 1973, pp. 331-74.

10. A.P. Prosser: Precious Metals "89, M.C. Jha and S.D. Hill, eds., TMS, Warrendale, PA, 1988, pp. 121~35.

11. J.C. Box and A.P. Prosser: Hydrometallurgy, 1986, vol. 16, pp. 77-92.

12. D.G. Dixon and J.L. Hendrix: Metall. Trans. B, 1993, vol. 24B, pp. 157-69.

13. M. Ishida and C.Y. Wen: AIChE J., 1968, vol. 14 (2), pp. 311-17.


14. R.B. Bird, W.E. Stewart, and E.N. Lightfoot: Transport Phenomena, John Wiley & Sons, New York, NY, 1960, pp. 73-74.

15. G.D. Smith: Numerical Solution of Partial Differential Equations, Oxford University Press, London, 1965, pp. 17-23.

16. S. Yakowitz and F. Szidarovszky: An Introduction to Numeri- cal Computations, Macmillan, New York, NY, 1989, pp. 197-214.

17. F. Habashi: Montana Bureau of Mines and Geology, Bulletin 59, Butte, MT, 1967.

18. R. Schuhmann: Trans. SME-AIME, 1960, vol. 241, pp. 22-25. 19. C.N. Satterfield: AIChE J., 1975, vol. 21 (2), pp. 209-28. 20. A. Bondi: Chem. Tech., 1971, Mar., p. 185.


A Mathematical Model for Heap Leaching of One or More Solid Reactants From Porous Ore Pellets

Documents

Transcript of A Mathematical Model for Heap Leaching of One or More Solid Reactants From Porous Ore Pellets