Hydrodinamica Heap Leaching Ubc

7/28/2019 Hydrodinamica Heap Leaching Ubc

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Investigative study into the hydrodynamics of heap leaching processes

Sylvie C. Bouffard and David G. Dixon1

Department of Metals and Materials Engineering

The University of British Columbia

309-6350 Stores Road

Vancouver, British Columbia, Canada

V6T 1Z4

1 Corresponding author: above address, [email protected]


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ABSTRACT

Three mathematical models (mixed side-pore and profile side-pore diffusion with uniform or

distributed pore length) are derived in dimensionless form to simulate the transport of solutes

through the flowing channels and the stagnant pores of an unsaturated heap. Model parameters

are determined from experimental tracer residence time distributions using the least square

minimization approach. It is shown that the residence time distribution curves display a long tail

resulting from the very slow mass transfer (or diffusion) into the 1-6 cm stagnant pores, which

take up 5 times more space than the flowing liquid. The very large coefficients of determination

(R2 > 0.99) confirm the validity of all models, and especially the profile side-pore diffusion

model with distributed pore length.

The effects of five factors (agglomeration, addition of binder, particle size, solution flow rate and

bed height) are examined. Data from experimental residence time distributions prove that the

advection time is directly proportional to the column height and inversely proportional to the

flow rate. The two estimated parameters (stagnant liquid holdup and pore length) are only

marginally affected by any change in crush size, agglomeration technique or operating

conditions. This in turn suggests that the model can predict and/or simulate the hydrodynamic

behavior in taller columns and possibly heaps.


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KEYWORDS

Hydrodynamic modeling, tracer tests, liquid holdup, heap leaching, agglomeration, advection,

diffusion, unsaturated flow, binder


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INTRODUCTION

There have been several attempts during the last twenty-five years to model heap and dump

leaching processes, by which metal values are extracted from ore particles and recovered from

the leaching solution percolating through the ore bed. Early models involved two-dimensional

governing equations to describe the diffusion and the convection of air into the pore spaces of the

heap/dump [1, 2]. It was widely held that the supply of oxygen to the acidophilic bacteria

colonizing the ore surfaces was the rate-limiting process. During the last ten years, attention has

shifted towards the leaching solution and the ore particles, as the installation of low-pressure

blowers has fast become standard practice in sulfide heap leaching, thus eliminating the oxygen

limitation. The flow and transport equations defining the rate at which solutes travel through the

bed interstices, exchange across phase boundaries and diffuse through water-filled pores, now

constitute the backbone of any heap leaching model. Phenomena such as leaching reactions,

bacterial attachment and growth, solute precipitation, gas/liquid mass transfer, heat generation,

and ore decrepitation and compaction can be incorporated, as required, as source terms into the

principal governing flow equations.

The hydrodynamics of single and multi-component solutes in porous media, as well as those of

trickle bed reactors in the chemical engineering industry, are very well documented in the

literature. Although heaps, packed towers and trickle bed reactors can all be portrayed as packed

bed reactors, the size of their packing material and their operating liquid and gas flow rates are

radically different. Hence, this wealth of information is of limited use to the heap modeler. Let

us thus examine more closely the approaches taken by Dixon and Hendrix [3], as well as

Snchez-Chacn and Lapidus [4], to model heap hydrodynamics.


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One of the two critical facets considered was the distribution of the leaching solution throughout

the heap and its relationship to the flow rate. A fraction of the leaching solution was assumed to

be held up in the pores of the particles. In a dry ore bed, this solution would progressively push

the air out of the smaller voids to form stagnant water pockets, thereby saturating the pores.

Although Dixon and Hendrix [3] assumed that this liquid fraction existed only in the pores of the

particles, Snchez-Chacn and Lapidus [4] accounted also for the partial wetting of the external

surface of the particles by a stagnant liquid film. Whether the water film is uniform or

contiguous, or whether the stagnant pockets exist in the pores of the particles and/or at their

interfaces, suffice it to say that the stagnant liquid fraction accounts typically for 17 to 32% of

bed void volume (or 7 to 13% of the total bed volume assuming a total bed porosity of 40%), as

measured in copper dumps and columns loaded with rocks 5 to 152 mm in size [5]. The physical

properties of the solution and the shape, size and wettability (i.e. contact angle) of the particles

have been found to have a significant effect on the stagnant liquid holdup, as shown

experimentally by Schlitt [6]. It is also worth noting that data measured in dumps are not

directly transferable to heaps because the two processes differ widely in their geometries

(rectangular vs. conical), stacking techniques (1-15 m in height vs. 60 m, agglomerates vs.

boulders, conveyor belt stacking vs. truck dumping) and solution management procedures.

When the smaller openings are flooded, the leaching solution finds its way through the larger

channels. This free draining solution, hereafter referred to as the flowing liquid, percolates as

rivulets and films across the surfaces of the particles. The flowing liquid holdup corresponds

roughly to the liquid collected during drainage, and depends primarily on the packing

characteristics, and to a lesser extent on (1) the fluid physical properties, (2) the liquid flow rate

and (3) the gas flow rate. A substantial change in the flowing holdup is observed when the flow


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rate is large enough such that air-filled voids begin to pinch off and flooding occurs. Column

tests reveal that the flowing liquid holdup increases from 0.017 to 0.025 m3 liquid/m3 bed over

the range of liquid flow rates from 6 to 3100 L/m2-h [6]. Flooding occurred at a flow rate of

about 4500 L/m2-h, corresponding to a flowing liquid holdup of 19.6%.

The aforementioned two authors [3, 4] assumed that the flowing liquid moved vertically as a

front through the bed (ideal plug flow behavior) at a constant velocity given by the product of the

superficial velocity (e.g. 5-15 L/m2-h for sulfidic refractory gold and copper sulfide ores) and the

degree of void saturation (assumed to be constant). From a theoretical standpoint, the velocity at

any given location can be calculated more accurately using Darcys law, which takes into

account the pressure gradient associated with the hydrostatic head (sum of gravitational and

capillary forces) and the local permeability. Changes in permeability are attributed to

decrepitation, weathering, compaction, segregation, precipitation and evaporation. The greater

accuracy obtained by developing a rigorous hydrodynamic model calling for local velocity

calculations must be weighed against a simpler, yet less rigorous, model with shorter

computation times. However, an assumption of constant velocity would be invalid for heaps

operated with rest/rinse cycles [7].

As pointed out by Roman and Bhappu [8] in their review of heap hydrology, a study of the

hydrodynamics restricted to the notions of solution holdup and relative volumes would be

incomplete without addressing the issue of solute transfer across boundaries and diffusion into

stagnant pores. In similar fields of research, models have been derived on the basis of Ficks law

of diffusion, or using the concept of stirred tank reactors, or even with stochastic techniques such

as random walk and Monte Carlo analysis. The aforementioned two authors [3, 4] have


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exploited the first approach. Dixon and Hendrix [3] have assumed a rapid mass transfer at the

liquid/liquid interface and a slow diffusion of the solute into the pores. The transfer rate was

given by the diffusional flux on the pore side of the interface. In addition to the hypotheses laid

out in Dixon and Hendrixs model, Snchez-Chacn and Lapidus [4] have also considered the

existence of film mass transfer at the liquid/solid interface given by a first order linear term

estimated from correlations pertaining to hydrodynamics in trickle bed reactors. Although the

agglomerates may be comparable in size to the packing material (Raschig rings, Berl saddles) of

trickle beds, the liquid and gas flow rates in such reactors are nevertheless orders of magnitude

higher than the conditions prevailing in heap leaching. Another common way of estimating

transfer and diffusion rates is to perform tracer tests. Regrettably, there exists no published study

relevant to heap leaching. Furthermore, those studies concerned with dump leaching are either

purely qualitative [5], or else the interpretation (published in ref. 9) of the tracer curves

(published in ref. 6) fails to address the issue of solute transport within the stagnant liquid.

What is lacking from the models developed previously is a set of reliable and exact parameters

that are representative of heap leach operations. The aim of this paper is to combine the most

important transport and transfer phenomena into a 1D mechanistic model of the heap leaching

process under forced air advection, and to evaluate its parameters from actual experimental tracer

curves. This study is divided into four parts. First, the bed porosity, the stagnant and flowing

liquid holdup, and the initial slumping of an ore bed are determined under several sets of

conditions. Second, a series of pulse and step tracer tests are conducted under different

experimental conditions to obtain the residence time distributions, from which the model

parameters sought are estimated using a least square minimization scheme. Third, the goodness-

of-fit of the three models proposed, coupled with their estimated parameters, is evaluated by


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comparing the numerical solution to the experimental residence time distribution. Finally, the

experimental conditions selected in the second part of this work are arranged in sets of factorial

experiments. A statistical analysis is carried out to determine the importance of agglomeration,

binder addition, particle size, solution flow rate and column height.

MODEL DEVELOPMENT

On the basis of the seven diffusion models compiled and illustrated by Roman and Bhappu [Fig.

5 of ref. 8], we have further developed three two-phase models (Figs. 1 to 3) to simulate the

shape of the column tracer breakthrough curves. These models were used by Stollenwerk and

Kipp [10] to simulate the transport of molybdate through a 30-cm long column packed with

sediment collected from an uncontaminated aquifer. A couple of years later Dixon et al. [11]

revisited these models to predict the effluent concentration response of reagents from spent heaps

under both fresh and recycled water rinsing.

The very large flat surface area of a heap, as compared to its narrow sloped sides, imposes a one-

dimensional geometry with boundary conditions only at the top. The tracer in the flowing liquid

is assumed to move as a front at a constant velocity (plug-flow behavior assumption). Solute

transfer takes place at the stagnant/flowing interface, and thus retards its transport to the bottom

of the heap. The stagnant phase is viewed as an array of pores of uniform or variable

(distributed) length, oriented normal to the convection channels. Microporosities within a

particle, macroporosities between particles making up an agglomerate, and porosities between

agglomerates are put end to end to represent a single pore. We make no distinction between the

water held in the open porosities of the agglomerates and at their interface because it is unknown

whether agglomerates swell or disintegrate upon contact with water. It is also assumed that the


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diffusion driving force from the flowing liquid is greater than the gradient between adjacent

pores at all times. Hence, there is no cross-talk between individual diffusion (or stagnant)

pathways. The tracer is non-reactive and undergoes neither sorption nor ion exchange.

Mixed Side-Pore Diffusion Model

The concept of first order mass transfer between the stagnant and flowing regions, which

mathematically simplifies the concept of Fickian diffusion in the stagnant areas, has often been

used to describe the physical non-equilibrium transport of solutes. In the mixed side-pore

diffusion (MSPD) model (Fig. 1), the solution concentration is assumed uniform throughout the

pore. The solute transfers at the flowing/stagnant interface only. The mass balance in the

stagnant pores, coupled with its initial condition, is given by:

( )sfv

s

s CCka

1

t

C=

I.C.: Cs(0) = 0 Eq. (1)

where Cs and Cfare the solute concentration in the stagnant and flowing phase, respectively,

normalized with respect to the input concentration, aV is the total exchange area per unit volume

of heap, kaV is the overall mass transfer coefficient at the interface, and s is the stagnant liquid

holdup. Similarly, the mass balance in the flowing phase is written:

I.C.1: Cf(z,0) = 0)CC(ka

z

Cu

t

C

sfvff

f

=

B.C.2: Cf(0,t) = 1

Eq. (2)

where u is the flowing liquid superficial velocity and fis the flowing liquid holdup. Let us

define the time of advection, ta,and two dimensionless variables:


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Z

z=

u

Z

t fa

= f

s

=

where Z is heap height, is a dimensionless depth variable, ta is the advection (or liquid mean

residence) time, and is the volume ratio of stagnant to flowing liquid. Eqs. 1 and 2, and their

respective conditions, are now rewritten thus:

( )sf

f

vs CC

ka

t

C=

I.C.: Cs(0) = 0 Eq. (3)

)CC(kaC

tt

Csf

f

vf

a

f =

+

1 I.C.1:

B.C.2:

Cf(,0)

Cf(0,t)

=

=

0

1

Eq. (4)

The MSPD model is entirely defined by three parameters (ta, and kav).

Profile Side-Pore Diffusion Model

In the profile side-pore diffusion (PSPD) model (Figs. 2 and 3), the concentration profile in the

pores is controlled by a diffusional transport mechanism. The transfer rate at the open end of the

pores is then given by the diffusional flux, which depends on the concentration gradient in the

stagnant phase at the interface. No transfer takes place at the closed end of the pores. The mass

balance along some reference diffusion pathway, coupled with its initial and boundary

conditions, is given by:


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+

=

x

C

x

n

x

CD

t

C s

2

s2

ss

I.C.1:

B.C.2:

B.C.3:

Cs(x,0)

Cs(X,t)

0xx

sC

=

=

=

=

0

Cf

0

Eq. (5)

where D is the diffusivity of the solute in the pores, and x is the relative position in the pores.

The constant n is a diffusion factor equal to 0 for linear diffusion, 1 for cylindrical diffusion or 2

for spherical diffusion. The mass balance in the liquid phase is written for a pore of uniform

length X:

Xx

ffff x

C

X

)1n(D

z

Cu

t

C

=

+

=

I.C.1:

B.C.2:

Cf(z,0)

Cf(0,t)

=

=

0

1

Eq. (6)

For pores of different lengths, we can define a dimensionless pore length, , which normalizes

the pore length, X, with respect to a reference pore length, X*. The selection of a pore length

distribution function is somewhat arbitrary in the absence of data from experiments designed

specifically to measure the actual distribution. The Gates-Gaudin-Schuhmann distribution

function is used in this work for modeling the pore length. The normalized distribution has a

single parameter, m, and is given by:

1

1

0

d1m

m

max

min

d)

(n = = Eq. (7)

where is the ratio of the pore length to the maximum pore length. A plot of the distribution

(Fig. 4) for values of m ranging from 0.01 to 100 shows that the distribution reduces to a


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monosize length for values of m smaller than 0.1 or larger than 10. Let us make use of the

dimensionless parameters defined for the MSPD model, and introduce the dimensionless position

in the pore, , and the time of diffusion into the pore, td:

X

x=

2maxs

2sd X

Dor

X

D

t

1= for uniform or variable pore length, respectively

Eqs. 5 and 6 are now rewritten with the dimensionless variables using , ta and td.

+

=

C

n

C

t

1

t

C s2

s2

2d

s

where = 1 for uniform pore length

I.C.1:

B.C.2:

B.C.3:

Cs(,0)

Cs(1,t)

0

sC

=

=

=

=

0

Cf

0

Eq. (8)

12

11

=+

=

f

d

f

a

f C

t

)n(

C

tt

C

I.C.1:

B.C.2:

Cf(

,0)

Cf(0,t)

=

=

0

1

Eq. (9)

For pores of uniform length, the dimensionless variable in Eq. (9) is set to 1. For a distribution

of pore length, Eq. (9) must be combined with Eq. (7) and integrated between = 0 and = 1,

thus:


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Uniform pore length:

1

s

d

f

a

f !C

t

"

)1n(#

C

t

1

t

C

=

+=

+

Variable pore length:

$

d!C

$

mt

"

)1n(#

C

t

1

t

C

1

s1

0

3m

d

f

a

f

+

=

+

=

I.C.1:

B.C.2:

Cf(,0)

Cf(0,t)

=

=

0

1

Eq. (10)

The PSPD model is thus characterized by three or four parameters (t a, , td and possibly m)

depending on whether the pores are of uniform or variable length.

EXPERIMENTAL METHODOLOGY

Physical and Chemical Properties of the Ore

About 1300 kg of sulfidic refractory gold ore were shipped to the University of British Columbia

(UBC) in the form of drill cores. The cores were crushed to less than 19 mm, coned and

quartered. X-ray diffraction, coupled with Rietveld analysis, revealed the bulk sample to contain

77.54% w/w quartz, 11.25% muscovite, 8.12% pyrite, 1.37% marcasite, 0.67% arsenopyrite and

1.05% kaolinite. Its bulk specific gravity and its apparent porosity were measured according to

the ASTM C20-87 and C127-88 standards, and were found to be 2454 kg/m3 and 2.68% v/v,

respectively. The particle size distribution (Fig. 5) was best fitted by a Rosin-Rammler

distribution (Eq. 11) with parameters q and d* equal to 1.64 and 8.47 mm, respectively.

*

q

d

d%where)

%

exp(1)%

(F == Eq. (11)


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Ore Preparation

A fraction of the bulk sample was loaded as is (non-agglomerated) in the columns. The second

portion was agglomerated with a 3 M H2SO4 solution in the presence or absence of Nalco 9704

binder. The agglomeration was carried out batchwise. A plastic bin was inserted into a 9 ft3

cement mixer without baffles, and 25 kg of the bulk sample were placed into the bin with about

800 mL of solution. When required, a known amount of binder was dissolved into the solution

prior to its addition to the non-agglomerated ore to yield a final binder dosage of 0.5 kg/t ore.

The cement mixer was tilted at 30 and rotated continuously for 30 min. The agglomerates had a

final acid dosage of 9.4 kg H2SO4/t ore and a moisture content of 3.2 % w/w before drying.

They were spread evenly on a plastic sheet and left to dry for one week. The third and final

fraction of the bulk sample was sieved through a 9.5 mm screen, and the undersize fraction was

used as is or agglomerated according to the procedure described above. The physical properties

(specific gravity, porosity, particle size distribution) of the dried agglomerates were not

measured because they tended to disintegrate upon contact with water or when shaken on

screens.

Experimental Apparatus

The experimental setup (Fig. 6) consisted of a PVC column of 25.4 cm inside diameter and of

variable height (either ~50 cm or ~180 cm). Each column was equipped with a shallow conical

bottom plate for drainage, on top of which rested a perforated plate (6.35 mm holes evenly spaced)

for gas distribution and ore support. A 5 mm layer of glass wool was placed on top of the

perforated plate to prevent the loss of fines. All short-bed columns were loaded with 45 kg of non-

agglomerated or agglomerated ore, while the tall columns were loaded with 135 kg. The initial


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height, Zo, was recorded, and the bed slumped to a final height, Z, soon after wetting. Layers of

4 cm hollow plastic balls were placed on top of the ore bed to ensure uniform distribution of

solution. The solution was pumped from a 20 L pail using a peristaltic pump to the top of the

column where it dripped onto an inverted 10 cm plastic funnel. A 30 cm shaft passing through

the funnel was rotated at 30 RPM. The device was positioned at such a height above the bed that

the liquid was evenly distributed without splashing on the wall of the column. Nitrogen was

injected at the bottom of the column underneath the perforated plate at a flow rate of 0.85 L/min,

as measured with a calibrated rotameter. All experiments were performed at room temperature

and atmospheric pressure.

Hydrology

The bed porosity (including the agglomerate porosity), h, is defined as the ratio of the volume of

solution pumped upward to fill the bed pore spaces to the volume of the empty column (product

of surface area and height after slumping). The flowing liquid holdup, f, corresponds to the

volume of solution drained after interruption of the flow. Even after 24 h an extremely small

quantity of liquid continued to issue from the column. Therefore, for practical reasons the

flowing liquid holdup was defined as the total volume of liquid collected after 24 h of drainage.

The stagnant liquid holdup (including the agglomerate porosity, s) was determined in two ways.

The first method consisted of measuring the volume of solution drained during 24 h from a

column whose pore spaces were totally filled with water. According to the second method, the

volume of stagnant water was given by the difference between the volume of solution pumped

into a dry column and the volume of solution collected at the bottom after the onset of steady

state flow. The effects of the liquid flow rate, the agglomeration, the particle size and the height


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on the bed properties were studied in a series of tests described in Table 1. The experimental

values of the flowing liquid holdup were used to evaluate the time of advection, t a, thus cutting

down the number of unknown parameters to two or three.

Tracer Studies

The remaining unknown hydrodynamic parameters (, kav, X or Xmax,and m) are deduced from

a series of residence time distribution measurements carried out in a column with countercurrent

flow of gas and liquid. Failed attempts at using nickel and bromide as tracers led to the selection

of a 3.41 M sodium nitrate solution (laboratory grade reagent salt dissolved in deionized water)

as a non-reactive tracer. In a first approximation, this tracer will serve as a proxy for each of the

species (ferric, ferrous, copper, zinc, sulfate ions, etc.) expected to be found in typical leaching

solutions. The diffusion coefficient of this strong 1: 1 electrolyte is given by:

1

3NONa3NaNO D

1

D

12D

+

+= Eq. (12)

where the diffusion coefficients of sodium and nitrate ions in water at infinite dilution are 1.33

10-9 and 1.90 10-9 m2/s, respectively, at 25C [12]. The diffusion coefficient of sodium nitrate

is thus estimated at 1.56 10-9 m2/s.

Before the step or pulse injection of the tracer, the acidic solution generated by the agglomerates

upon contact with tap water was eluted. A 0.05 M NaNO3 solution was then pumped in until the

electrical conductivity measured by a conductivity flow-cell (Model 1481-65, Cole-Parmer)

located downstream of the bottom drainage port was constant. The pulse injection then occurred

by manually taking out the inlet tubing from the 0.05 M solution pail and inserting it into the


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3.41 M NaNO3 solution flask. The injection was complete in less than 45 min, after which the

inlet tube was wiped and re-immersed into the 0.05 M solution. The signal of the conductivity

meter was amplified and transmitted to a computer by a data acquisition card (Keithley

Metrabyte, MA) and a data acquisition system (Labtech Notebook, MA). The computer

recording was synchronized with the introduction of each pulse or step. It was unnecessary to

convert the conductivity readings into concentration values because there exists a linear

relationship between the two over the range of concentrations tested. In total fourteen tests, each

characterized by a unique set of experimental conditions, were conducted to study the effects of

the liquid flow rate, the agglomeration, the addition of binder, the particle size and the height

(Table 2).

NUMERICAL SIMULATIONS

Analysis of the Experimental Data

The conductivity measurements for a step input are easily converted to dimensionless

concentrations ( = C/Cfinal) to yield the so-called F curve. For a pulse input, the conductivity

measurements are first divided by the area under the concentration-time curve to generate the

normalized C concentrations. The F values are then calculated by integrating the area under the

C-time curve from 0 to the time at which C is detected. The integration is performed using the

simple trapezoid method. In short, the C curve corresponds to the derivative of the F curve, and

accordingly the F curve is the integral of the C curve.

Numerical Solution

The equations corresponding to the MSPD and PSPD models are transformed into ordinary

differential equations through the definition of the substantial rate derivative [13], expressed in


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dimensionless coordinates:

&

tt&

D

D

tDt

D

aa

+

==11

Eq. (13)

A unit analysis reveals that the left- and right-hand side of Eq. 12 have both units of inverse time.

This transformation facilitates the mathematical treatment of the model from the Laplacian

viewpoint, i.e. via the tracking of fluid elements through the column. For this, a simple

backward finite difference approximation is used [14]:

)CC(&

'

1(

D

DC

j,1kj,k = Eq. (14)

where k is the column depth index and j is the time index. The integral on the right-hand side of

Eq. 10 is solved with the Gauss-Legendre quadrature. The order of solution of the model

equations is as follows: (1) all initial conditions are defined, (2) starting at time t = t a and heap

depth = 0, the concentrations Cs are determined using a fully implicit scheme in solving the

second-order parabolic differential equation, (3) the pore entrance concentration gradient is

calculated, and this and the pore entrance concentration are used to evaluate Cfat + . Steps

(2) and (3) are repeated for each depth increment until = 1. The procedure is then repeated for

the new time, t + t, using the new set of Cs concentrations. The algorithms code is written

using the software Microsoft FORTRAN Visual Workbench version 1.0.

The model parameters are estimated by the least square minimization technique. In this work, a

multidimensional unconstrained nonlinear minimization routine is written to find the minimum

value of the sum of squares of the residuals, SSE, by adjusting simultaneously two or three


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parameters (s andkaV, s and X,or s, Xmax and m). The coefficient of determination, R2, is

calculated to assess the adequacy of the fit (Eq. 15).

( )n

SandSSwhereS

SS1R

2

n

1iin

1i

2i

n

1i

2

iiEFF

E2

= == ===

F

FFF FF Eq. (15)

STATISTICAL ANALYSIS

In this work, the effects of five factors (agglomeration, binder addition, particle size before

agglomeration, solution flow rate and bed height) are investigated in fourteen random tests

(Table 3). Up to four replicates are tested for a given set of conditions. Not all replicates are

perfectly genuine because of the time-consuming and labor-intensive nature of the

loading/unloading and clean-up process in between tests. The five factors are arranged in six

pairs: agglomeration/size (pair 1), binder/size (pair 2), agglomeration/flow (pair 3), flow/size

(pair 4), height/flow (pair 5) and binder/flow (pair 6). Each factor is tested at two levels, i.e.

agglomerated or non-agglomerated, with or without binder, slow or fast flow, small or large

particles, and short or tall bed. This factorial arrangement generates four sets of experimental

conditions for each pair. The four means of the stagnant liquid holdup and of the pore length are

then compared to one another.

RESULTS and DISCUSSION

The characteristics of the experiments and their corresponding model parameters are presented in

Tables 1 to 3. Before commenting on the significance of the bed properties and model


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parameters, let us discuss the shape of the F andC curves and compare the model fits to the

experimental data.

Characteristics of the Tracer Curves

The moderately asymmetrical shape of the experimental step curves (Figs. 7a and 7b) and pulse

curves (Fig. 8a and 8b), as well as the noticeably flat tail, suggest the presence of an important

stagnant fluid phase. The breakthrough is not instantaneous; the tracer concentration in the

effluent increases progressively, leading us to believe that the exchange between the stagnant and

flowing phases, or likewise the diffusion throughout the stagnant pores, is slow.

On a C- or F-curve plot, the advection time corresponds to the time at which the first non-zero

concentration is detected. In light of our assumption of plug-flow behavior, the advection time

corresponds to the minimum time a tracer molecule in the flowing liquid would spend in the

column before being detected in the effluent. This definition is strictly valid if (1) the transfer in

the stagnant pores is slow and (2) the flowing liquid phase moves down as a front at a constant

velocity. By making use of the definition of ta and the data presented in Table 2, we calculate an

advection time of 1.4, 3 and 6 h for the three tests presented in Fig. 7a. Graphically, we find

values of 2, 5 and 10 h, which are roughly 1.6 times larger than the calculated ones. But most

interesting of all is the relationship between the advection time and the two most important

operating conditions, i.e. the flow rate and the column height. According to the definition, the

advection time is directly proportional to the height and inversely proportional to the velocity.

Increasing the height from 0.58 to 1.64 m (a factor of 2.82) yields an advection time roughly 2.5

times larger (Figs. 7a and 7b). Doubling the flow rate has the exact opposite effect.


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21

MSPD vs. PSPD Models

For the purpose of comparing the goodness-of-fit of the MSPD and PSPD models, we have fitted

the tracer concentration in the effluent following a step input in a 0.52 m (test 2, best case

scenario) and 1.64 m (test 10, worst case scenario) tall column under the conditions given in

Table 2. The three coefficients of determination are evaluated for the specific case whereby: (1)

the heap is partitioned into 75 depth intervals, (2) each stagnant pore is separated into 20 regions

(the MSPD model does not call for such a condition), and (3) the test duration is divided into 10

min intervals. Inspection of Fig. 9a reveals that the fits of the three models match perfectly the

experimental F concentrations, with minor deviations of the MSPD model fits observed near the

breakthrough point (< 10% F value) and near complete saturation of the pores with sodium

nitrate (> 90% F value). The deviations of the MSPD model fits are amplified in the 1.64 m

column (Fig. 9b). Despite the shorter computational time the MSPD model does not fit the

experimental data as well as either of the two PSPD models (Table 3, Figs. 9a and 9b). In the

context of actual heap leaching experiments whereby ions interact with solid minerals and with

each other, the PSPD model may best describe the change in solute concentration in the stagnant

pores as a result of transport/transfer phenomena coupled with one or more chemical reactions

occurring at the grain boundaries.

Although the PSPD model with uniform pore length describes remarkably well the data of all

short column experiments (Table 2), the fit of the tracer curves of the tall columns is not as good

as that of the PSPD model with variable pore length (Figs. 7a vs. 7b, Figs. 8a vs. 8b, and Figs. 9a

vs. 9b). We note, indeed, a greater discrepancy between the fitted and experimental points

beyond 30 h (Fig. 9b) for the PSPD model with uniform pore length. The three-parameter


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22

model, on the other hand, yields a perfect fit of all experimental F data (Fig. 9b) no matter what

type of diffusion (either spherical, cylindrical or linear) (Table 3). More noteworthy is the much

improved fit of the C concentration data with the pore length distribution parameter, m. The fit

is truly remarkable. The addition of a third parameter in the optimization routine minimized the

error and yielded coefficients of determination of 0.9999, suggesting a near-perfect fit. In a word

the excellent match of the fitted points and experimental data confirms the validity of the PSPD

model.

One could argue that any fit will always improve with a larger number of estimated parameters,

which is generally true. Regardless of our experimental measurements of the stagnant liquid

holdup, the s variable was always one of the two or three fitted parameters during the

minimization search, and thus was never assigned to any of the experimental data. In some

cases, we found that the fitted s values were about twice as large as the experimental ones

(Table 1), suggesting, at first glance, lack of agreement between experimental and fitted data.

However, even under similar experimental conditions, the measured s values differed in fact by

as much as two-fold (Table 1). On that basis, we believe that the three-parameter PSPD model

could be simplified to a two-parameter PSPD model where Xmax and m would be the only two

fitted parameters. Besides, s values can be easily measured in column tests.

Type of Diffusion

We have considered spherical, cylindrical and linear diffusion to model the data of the tall

column experiments using the PSPD model with uniform or variable pore length. One would

expect the spherical type of diffusion to describe more adequately the diffusion into the more or

less spherical agglomerates. Table 3 shows indeed that the spherical type yields the best fit,


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followed by the cylindrical and linear types. The pore length is also directly proportional to the

diffusion factor (either 0, 1 or 2). This trend can be explained by considering the volume of a

sphere, a cylinder and a slab. Let us consider, for instance, a sphere with a radius of 6 cm. Its

total volume is thus equal to 905 cm3. For the purpose of this example we assume also that this

entire volume corresponds to a stagnant zone filled with solution. Because this parameter is

constant for all types of diffusion, we can hypothesize the volume of 905 cm3 to be constant for

the cylinder and the slab. We will set the thickness of the cylinder and the depth and length of

the slab to be equal to the diameter of the sphere. For the volume of the cylinder and of the slab

to be 905 cm

3

, the radius of the cylinder and the width of the slab must be 4.89 and 3.14 cm,

respectively. We note a decrease of the pore length from 6.00 to 4.89 to 3.14 cm as the geometry

changes from a sphere to a cylinder to a slab. That is why the same trend is observed in Table 3.

Despite the variation amongst the coefficients of determination, the theoretical ratios

Xspherical/Xcylindrical (6.00/4.89 = 1.27), Xcylindrical/Xlinear (4.89/3.14 = 1.56) and Xspherical/Xlinear

(6.00/3.14 = 1.91) are still in fair agreement with the predicted ones (1.30-1.38, 1.5-1.7, and 1.9-

2.33, respectively) (Table 3).

Bed Void Fraction

Immediately after a column was packed with dry agglomerates, the pump was turned on and the

liquid was dispersed. The initial bed height decreased by about 5 cm just a few hours later. This

corresponds to a reduction of 1.1 to 10.2% (h/h) depending on the packing material (Table 1).

Columns loaded with agglomerates experienced a slump 2 to 5 times larger than those packed

with non-agglomerated material. This is expected since dry agglomerates are more spherical and

more uniform in size than non-agglomerated ore. Both of these factors lead to a higher void

fraction before wetting [15]. The bed after slumping had an average void fraction of 43% (Table


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1), a value in good agreement with results previously reported in the literature on packed bed

reactors.

Stagnant Liquid Holdup

When the liquid flow was at steady state, approximately 15% (experimental range from 7 to

23%) of the bed was filled with stagnant water (Table 1). The PSPD model predicts a stagnant

liquid fraction of 12.6 4.4% (95% confidence level) (Table 2). The experimental results, the

estimated values and the data available in the literature (7-13%) [5] are all in very good

agreement. The analysis of the means reveals that, among all factors, the ore particle size before

agglomeration has the most significant effect on the fraction of liquid held up in pores. Smaller

agglomerates and/or particles have more surface area exposed to the stagnant liquid. As shown

in Table 4 (pairs 1, 2 and 4), the finer the crush size, the larger the stagnant holdup.

Parameter

The experimental values (2.1 to 3.3%, average 2.8%) of the flowing liquid holdup determined

after 24 h of drainage are in excellent agreement with the literature data (1.7-2.5%) [6]. Using

these values we find that the experimental ratio (6.07 4.91) of the stagnant to flowing liquid

holdup agrees very well with the predicted ratio, , of 5.00 1.80. Let us now consider the

physical meaning of this parameter. First, the solute is more likely to diffuse through the

numerous stagnant water pockets than to be dispersed into the flowing liquid. Therefore, our

assumption that axial dispersion phenomena could be ignored was fully justified. Second, we

would expect if a step input were applied to a column containing a large volume of stagnant

liquid that the diffusion time would be considerably longer than the advection time. The

solute molecules would be more spread out into the long pores. Correspondingly, a smaller


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value of implies that much of the solute would remain in the flowing liquid and exit rapidly.

The effect of on the shape of the F curve is studied with the PSPD model using values of 1,

2, 5, 7 and 10 while maintaining every other parameter constant. It can be clearly seen from

Fig. 10 that the shape of the F curve progressively shifts from an elongated S-shape to a more

rectangular wave (i.e. a step) with decreasing values. Interestingly, even when both phases are

present in equal proportion ( = 1), the F curve looks very much like a step.

Mass Transfer Coefficient

The estimated values of the parameter kaV for the 1.64 m tall column range from 5.8 10-6 to

11.5 10-6 s-1. In contrast, the same mass transfer coefficient (0.1 - 1.2 s-1) in downflow and

upflow trickle bed reactors is about five orders of magnitude larger [16]. Two parameters are

incorporated in kaV: the overall mass transfer coefficient, k, and the interfacial area, aV. Let us

first focus our attention on the effect of the velocity on k. According to the correlation proposed

by Wilson and Geankoplis [17] for the mass transfer coefficient at the flowing/stagnant interface

in saturated systems, and later modified by Gierke et al. [18] for unsaturated systems, k is

proportional to the velocity to the 1/3 power at low Re numbers. In our experiments (Table 3),

the ratio of k2u/ku is 1.90 compared to the theoretical value of 1.26. Because the original

correlation [17] was proposed for saturated packed bed reactors with a total porosity ranging

from 35 to 75%, using it for predicting the mass transfer coefficient might be unadvisable given

that, in our experiments, the flowing liquid holdup accounted for only 3% of the total column.

We note, however, a fair agreement between the experimental and theoretical ratios when the

correlation as a tool in a sensitivity analysis of the velocity.


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26

Pore Length

Results from twenty-two simulations indicate that the uniform pore length of the two-parameter

PSPD model ranges from 1.46 to 6.78 cm, with an average of 3.8 2.5 cm (95% confidence

interval) (Table 2). When pores are assumed to be of variable length, very few pores could be as

long as 15 cm (Table 3). Let us not forget, however, that the pore length distribution function of

the three-parameter PSDP model is defined by the maximum pore length (Xmax = 15 cm) and the

size distribution parameter, m. The latter ranges from 0.58 to 1.98, suggesting a uniformly

weighed distribution with as many short pores as long ones (refer to Fig. 4). The average value

of the distribution corresponds to the ratio of m over (m+1). Needless to say that the average

pore length of the three-parameter PSPD model is very similar to the uniform pore length of the

two-parameter model (Table 3).

When considering exclusively the diffusion into a particle with tortuous pores distributed

throughout their volume, Epstein [19] has demonstrated that the tortuosity was roughly

proportional to the square root of the actual pore length to the radius of the particle. In our

experiments, the largest agglomerate was about 19 mm in size and the average pore was roughly

38 mm long. Taking the tortuosity factor, , as 3.0, we evaluate the radius of an agglomerate

to be about 22 mm. Clearly the pores do not represent the actual tortuous path within an

agglomerate. Nevertheless the pore length seems to be directly proportional to the particle size

(Table 4). Indeed the same solute would necessarily take more time to diffuse throughout a large

particle than a smaller one. Similarly, when fine particles stick to one another to form

agglomerates, they no longer exist as a multitude of single small particles but rather as large

ones. That is why pores in a column packed with agglomerated ore seem longer than those in a


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column loaded with non-agglomerated particles (Table 4).

The limited number of tests carried out in the 1.64 m column indicate that the pore length

increases by only 1.55 cm as compared to the shorter 0.5-0.6 m columns (Table 4). One possible

explanation for the increasing pore length could be the coalescence of small flow channels into

larger ones, thus leading to channeling [20]. To convince us, nonetheless, that this trend is not

statistically significant at a 95% confidence limit, we can compare this 1.55 cm increase to the

2.50 cm standard deviation of the mean of the pore length. Clearly the difference is well within

the 95% confidence interval. And also let us not forget that three tests only were performed in

the taller column. We can say confidently that the PSPD model and its two parameters is

applicable to a short column just as its predictions fit very nicely the tracer distribution of a taller

column. In short the model can be scaled up with height.

Table 4 indicates also that the pores get shorter with increasing velocities. Perhaps the effective

distance between flow channels gets smaller at higher flow rate, or else, the open-end of the

stagnant pores is set in motion. Then again the average diminution is about 1.2 cm. We could

argue on the basis of the standard deviation that this trend is also not statistically significant, and

for good reasons considering that the two flow rates tested are well below the saturation criterion

and only differ from one another by a factor of two.

In light of this discussion let us now revisit the definition of the advection time and the diffusion

time. Based on the flowing liquid holdup, the column height and the flow rate, we calculate an

average advection time of 2.7 h for all 22 tests performed. Similarly, the diffusion time is found

to be about 35.4 h, which is approximately 13 times larger than the advection time. It is not

surprising therefore that the tracer curves exhibit a long tail. But what would be the effect on the


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28

shape of the tracer curve if the pore length were to increase or decrease while all other variables

were to remain the same? To answer this question let us consider Fig. 11 depicting five curves

with values of X ranging from 1 to 9 cm for conditions typical of our column experiments. In

contrast to the curves shown in Fig. 10, all curves depicted in Fig. 11 cross each other in the

neighborhood ofF ~ 0.6-0.7. The shorter the pore length, the longer the time before the

appearance of the solute at the bottom of the column. This suggests that, for smaller X values,

the diffusion (or the interfacial transfer) is very fast such that all stagnant pores are totally

saturated before breakthrough. Under these conditions, the stagnant and flowing phases make up

a single phase, which in turn yield an F curve characteristic of the quasi-ideal plug-flow

behavior. The curve stretches gradually to a more elongated S-shape with increasing pore

length. This suggests that part of the solute in the flowing liquid makes it through the entire

column before the remaining solute has time to diffuse to the end of the pores.

The remarkable fits obtained with the three models developed herein challenge a comment made

by Roman and Bhappu [8] about the inadequacy of the diffusion models to accurately predict

the residence time distribution of a solute within the pores spaces of a heap. The three models

may not be perfect analogs of the actual heap hydrodynamics, but as Levenspiel said, in many

cases we really do not need to know very much, simply how long the individual molecules stay

in the vessel, or more precisely, the distribution of residence times [21].

CONCLUSIONS

This in-depth investigation into the hydrodynamics of heap leaching reveals that the major

transport and transfer phenomena can be described remarkably well by the simple


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advection/mass transfer model or the slightly more complex advection/diffusion model. The

void volume of a column packed with agglomerated or non-agglomerated ore is roughly 43%.

Of that void space 28 to 47% is filled with water, the rest occupied by air. The liquid phase is

partitioned into (1) a stagnant liquid held in pores 1-6 cm in length and (2) a liquid phase flowing

through the larger interstices. The ratio of stagnant to flowing liquid is approximately 5 to 6. A

comparison of the experimental and estimated values of the stagnant liquid holdup suggests that

the original two-parameter profile side-pore diffusion model can be simplified to a one-

parameter model wherein the sole unknown parameter would be the pore length, X. The long

tail of theC

tracer curve indicates that the solute exchange at the interface of the two phases, or

the transfer across the pores, is very slow.

Of the five factors tested (agglomeration, binder addition, particle size, liquid flow rate and bed

height), the effects of the velocity and of the height were the most significant. Increasing the

velocity and/or decreasing the column height somewhat shorten the stagnant pores. Their effect

on the advection time is much more significant and totally predictable. The stagnant liquid

holdup seems to decrease very slightly whether the ore is agglomerated with binder or crushed to

a smaller size.

In light of the previous publications pertaining to the hydrology of heap leaching, this is without

any doubt the most comprehensive study of its kind. Notwithstanding the excellent fit between

the predicted and experimental curves, one could question the validity of our assumptions on a

larger scale where might exist distinct wet and dry zones, as well as preferential flow channels.

Similarly, whether the model would still be applicable throughout the entire duration of the leach

(possibly longer than one year) during which (1) the ore particles may disintegrate, (2) the fines


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may migrate towards the bottom of the heap, and (3) the precipitates may plug off some pores,

remains to be seen? Remember that, according to the Rosin-Rammler particle size distribution,

our original bulk sample contained roughly equal amounts of particles (1) passing 3.2 mm, (2)

+3.2-6.4 mm, (3) +6.4-9.5 mm, and (4) +9.5-12.5 mm. How would the model parameters

change if the experimenter purposely increased the fines content to expose more reacting

surfaces to the leaching solution? Should the modeler also be concerned with the interfacial

phenomena, such as the adsorption/desorption of leaching reactants and products onto the

agglomerates? These are all valid issues that could be addressed in future experimental work.

Furthermore, the models assume that pore diffusion within the mineral particles is lumped with

the diffusion through the agglomerate intraporosities and interporosities. If pore diffusion

through the already-existing porosities of the mineral particles was rate-limiting, it could be

coupled with the models using an appropriate form of a microscale particle leaching model

(diffusion-controlled shrinking-core or grain-scale model). For the most part intraparticle

porosities are created as leaching progresses, i.e. during the dissolution of gangue minerals in the

presence of sulfuric acid or the oxidation of sulfide veins. Let us not forget, however, that

sulfide and oxide grains residing on the external surfaces of the ore particles would be exposed to

a higher concentration of reagents than those buried deep down within the ore pellets, and would

thus leach faster.

Regardless of these considerations, the profile side-pore diffusion model and the parameters

estimated in this work provide a more than adequate fit of the tracer data. In the absence of any

meaningful experimental data pertaining to the hydrology and hydrodynamics of heaps, the

model developed herein establishes the foundations for a more advanced model of the heap


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31

leaching process.

ACKNOWLEDGMENT

The work was supported by Placer Dome Inc., the Natural Sciences and Engineering Research

Council of Canada, and the CRD Grant 233710. The authors gratefully acknowledge the

technical support of Masoud Aftaita, Elisabetta Pani, Shawn Crego and the staff of the UBC

Coal and Mining Processing Laboratory.


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stagnant to flowing holdup ratio

dimensionless pore length

normalized dimensionless pore position

dimensionless depth position

dimensionless tortuosity factor

SUBSCRIPTS

h heap

f flowing liquid

s stagnant liquid

REFERENCES

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

2. G.B. Davis and A.I.M. Ritchie: Applied Math. & Modelling, 1986, vol. 10, pp. 314-22.

3. D.G. Dixon and J.L. Hendrix: Metall. Trans. B, 1993, vol. 24B, pp. 1087-02.

4. A.E. Snchez-Chacn and G.T. Lapidus: Hydrometallurgy, 1997, vol. 44, pp. 1-20.

5. L.E. Murr, W.J. Schlitt and L.M. Cathles: Experimental observations of solution flow in the

leaching of copper-bearing waste. Interfacing Technologies in Solution Mining.

Proceedings of the Second SME-SPE International Solution Mining Symposium, W.J. Schlitt

and J.B. Hiskey, Denver, CO, 1981, pp. 271-290.

6. W.J. Schlitt: The role of solution management in heap and dump leaching. Au and Ag Heap

and Dump Leaching Practice, J.B. Hiskey [Ed.], SME-AIME, NY, 1984, pp. 69-83.

7. J. Martinez, L. Moreno and J.M. Casas: Modeling of intermittent irrigation in leaching heaps.

Copper International Conference, vol. 3: Electrorefining and Hydrometallurgy of Copper,


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CIM, 1995, pp. 393-408.

8. R.J. Roman and R.B. Bhappu: Heap hydrology and the decommissioning of a leach heap.

Hydrometallurgy, Fundamentals, Technology and Innovation. Proceedings of the Milton E.

Wadsworth (IV) International Symposium on Hydrometallurgy, J.B. Hiskey and G.W.

Warren, Littleton, CO, 1993, pp. 1061-72.

9. R.W. Bartlett: Metall. Trans. B, 1997, vol. 28B, pp. 529-45.

10. K.G. Stollenwerk and K.L. Kipp: Simulation of molybdate transport with different rate-

controlled mechanisms. ACS Symposium Series: Chemical Modeling of Aqueous Systems

II, American Chemical Society, DC, 1990, pp. 241-57.

11. D.G. Dixon, R.B. Dix and P.G. Comba: Extraction and Processing for the Treatment and

Minimization of Wastes, J. Hager, B. Hansen, W. Imrie, J. Pusatori and V. Ramachandran

[Eds], The Minerals, Metals & Materials Society, 1993, pp. 701-13.

12. E.L. Cussler: Diffusion Mass Transfer in Fluid Systems, 2nd ed., Cambridge University

Press, Cambridge, United Kingdom, 1997, pp. 142-48.

13. R.B. Bird, W.E. Stewart and E.N. Lightfoot: Transport Phenomena, John Wiley & Sons,

New York, NY, 1960, pp. 73-74.

14. G.D. Smith: Numerical Solution of Partial Differential Equations, Oxford University Press,

London, 1965, pp. 17-23.

15. W.L. McCabe, J.C. Smith and P. Harriott: Unit Operations of Chemical Engineering, 5th ed.,

McGraw-Hill, Inc., New York, NY, 1993, pp. 154-55.


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16. I. Iliuta, F.C. Thyrion, L. Bolle and M. Giot: Chem. Eng. Technol., 1997, vol. 20, pp. 171-81.

17. E.J. Wilson and C.J. Geankoplis: Ind. Eng. Chem. Fundam., 1966, vol. 5, pp. 9-14.

18. J.S. Gierke, N.J. Hutzler and J.C. Crittendon: Water Resour. Res., 1990, vol. 26, pp. 1529-47.

19. N. Epstein: Chem. Eng. Sci., 1989, vol. 44, pp. 777-79.

20. G.A. Funk: Ind. Eng. Chem. Res., 1990, vol. 29, pp. 738-48.

21. O. Levenspiel: Chemical Reaction Engineering, 2nd ed., John Wiley & Sons, New York, NY,

1972, p. 254.


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LIST OF FIGURES

Figure 1. Representation of the MSPD model.

Figure 2. Representation of the PSPD model with uniform pore length.

Figure 3. Representation of the PSPD model with variable pore length.

Figure 4. Gates-Gaudin-Schuhmann distribution for values of m ranging from 0.01 to 100.

Figure 5. Ore particle size distribution fitted with the Rosin-Rammler function.

Figure 6. Experimental apparatus.

Figure 7a. Comparison of experimental and fitted F concentration for short and tall columns

(PSPD model with uniform pore length).

Figure 7b. Comparison of experimental and fitted F concentration for short and tall columns

(PSPD model with variable pore length).

Figure 8a. Comparison of experimental and fitted C concentration for short and tall columns


Figure 8b. Comparison of experimental and fitted C concentration for short and tall columns


Figure 9a. Comparison of the MSPD and PSPD model fits for test 2 (best case scenario).

Figure 9b. Comparison of the MSPD and PSPD model fits for test 10b (worst case scenario).

Figure 10. Effect of at constant X. Spherical type of diffusion (n = 2).

Figure 11. Effect of X at constant . Spherical type of diffusion (n = 2).


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37

z=0

z=Z

x=X x=0

Stagnant

pores

Flowing

liquid

u

Injection

Measurement

Figure 1. Representation of the MSPD model.


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38

z=0

z=Z

x=X x=0

Stagnant poresFlowing

liquid

u

Injection

Measurement

Figure 2. Representation of the PSPD model with uniform pore length.


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39

z=0

z=Z

Flowing

liquid

u

Injection

Measurement

Stagnant pores

Figure 3. Representation of the PSPD model with variable pore length. The total area of

each group of bars is equal to the area of each dashed box in the stagnant region.


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40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Pore length, (dimensionless)

F()

m = 0.01

m = 0.1

m = 1

m = 10

m = 100

Figure 4. Gates-Gaudin-Schuhmann distribution for values of m ranging from 0.01 to 100.


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41

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Particle size, d (mm)

F()

Figure 5. Ore particle size distribution fitted with the Rosin-Rammler function.


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42

Figure 6. Experimental apparatus. 1)influent, 2) conductivity cell, 3) effluent, 4) data

acquisition system, 5) ore, 6) rotameter, 7) perforated plate, 8) glass wool, 9)

plastic balls, 10) inverted funnel, 11) peristaltic pump with controller,

12) rotating shaft, 13) stirrer motor and 14) motor controller.

N2, 25 psig

1

2

4

5

6

7

8

9

10

11

12

1314

3


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43

Figure 7a. Comparison of experimental and fitted F concentration for short and tall columns



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44

Figure 7b. Comparison of experimental and fitted F concentration for short and tall columns



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45

Figure 8a. Comparison of experimental and fitted C concentration for short and tall

columns (PSPD model with uniform pore length).


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46

Figure 8b. Comparison of experimental and fitted C concentration for short and tall

columns (PSPD model with variable pore length).


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47

Figure 9a. Comparison of the MSPD and PSPD model fits for test 2 (best case scenario).


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Figure 9b. Comparison of the MSPD and PSPD model fits for test 10b (worst case scenario).


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Figure 10. Effect of at constant X. Spherical type of diffusion (n = 2).


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Figure 11. Effect of X at constant . Spherical type of diffusion (n = 2). Pore length, X, has

units of cm.


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51

LIST OF TABLES

Table 1. Experimental conditions and results of hydrology tests.

Table 2. Estimated parameters and goodness-of-fit of the PSPD model (spherical type of

diffusion only) for all short and tall column experiments.

Table 3. MSPD and PSPD estimated parameters and goodness-of-fit for the three tall

column experiments.

Table 4. Comparison of the means of six two-factor sets.


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52

Test

Ore*

Weight

(kg)

Binder

ParticleSize

(mm)

Slu

mp

(%

h/h)

u

(103m

/h)

h*

*

(m3 voids/m3bed)

s

(m3 liquid/m3 bed)

f

(m3 liquid/m3 bed)

a

NA

45

---

9.5

2

.1

5.08

NM

0.23

0.032

b

NA

45

---

9.5

2

.1

10.15

NM

0.23

0.032

c

A

45

No

9.5

6

.5

5.05

NM

0.15

0.031

d

A

45

No

9.5

6

.5

9.95

NM

0.15

0.027

e

A

45

Yes

9.5

6

.5

---

NM

NM

NM

f

NA

45

---

19.1

1.1-2.8

5.08

0.43

0.07-0.21

0.027-0.033

g

NA

45

---

19.1

1.1-2.8

10.15

0.43

0.07-0.21

0.025

h

A

45

No

19.1

4.2-

10.2

4.90

0.43

0.10

0.021

i

A

45

No

19.1

4.2-

10.2

10.01

0.43

0.10

0.024

j

A

137

No

19.1

11.0

5.25

NM

0.22

0.018

k

A

137

No

19.1

11.0

10.2

NM

0.22

0.021

l

A

45

Yes

19.1

1

.1

---

NM

NM

NM

* A:agglomerated,NA

:non-agglomerated

**N

M:notmeasured

Table1.

Experimentalconditionsandresultsofhydrologytests.


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53

Test Tracer*

Ore BinderP.S.**

(mm)

u

(106 m/s)

Z

(m)Data

f(exp.)

s(exp.)

s(est.)

X

(cm)R

2

1 P NA --- < 9.5 1.43 0.521 618 0.032 0.23 0.1709 2.605 0.9993

2 P NA --- < 9.5 2.83 0.521 202 0.032 0.23 0.1715 1.455 0.9999

3 P NA --- < 19.1 1.43 0.584 520 0.030 0.14 0.1170 3.964 0.9997

4 P NA --- < 19.1 2.83 0.584 210 0.025 0.14 0.1094 2.764 0.9997

5a P A No < 9.5 1.41 0.552 529 0.031 0.15 0.1093 3.443 0.9999

5b S A No < 9.5 1.41 0.552 472 0.031 0.15 0.1122 3.610 0.9993

6a P A No < 9.5 2.76 0.552 399 0.027 0.15 0.1245 2.931 0.9997

6b P A No < 9.5 2.76 0.552 303 0.027 0.15 0.1255 3.034 0.9996

7a S A No < 19.1 1.43 0.559 114 0.021 0.11 0.1048 3.883 0.9977

7b S A No < 19.1 1.38 0.584 190 0.021 0.10 0.1409 6.784 0.9899

7c P A No < 19.1 1.43 0.559 99 0.021 0.10 0.1150 4.635 0.9997

7d P A No < 19.1 1.38 0.584 528 0.021 0.10 0.1256 4.872 0.9978

8 P A No < 19.1 1.37 1.638 1681 0.018 0.22 0.1140 6.686 0.9963

9 P A No < 19.1 2.70 0.584 270 0.024 0.10 0.1173 3.303 0.9998

10a S A No < 19.1 2.80 1.638 677 0.019 0.22 0.0911 4.656 0.9996

10b P A No < 19.1 2.80 1.638 915 0.019 0.22 0.1116 4.928 0.9976

11a S A Yes < 9.5 1.46 0.572 434 0.031 0.15 0.1203 3.417 0.9987

11b P A Yes < 9.5 1.46 0.572 364 0.031 0.15 0.1301 3.300 0.9998

11c P A Yes < 9.5 1.45 0.533 638 0.031 0.15 0.1514 4.396 0.9994

12 S A Yes < 9.5 1.98 0.533 481 0.027 0.15 0.1706 2.965 0.9991

13 P A Yes < 19.1 1.32 0.61 657 0.021 0.10 0.1182 3.753 0.9996

14 P A Yes < 19.1 2.48 0.61 257 0.024 0.10 0.1154 3.018 0.9986

* P: pulse, S: step

** P.S.: particle size

Table 2. Estimated parameters and goodness-of-fit of the PSPD model (spherical type of

diffusion only) for all short and tall column experiments.


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54

Test 8 Test 10a Test 10b

U 4.97 10.08 10.08

f 0.018 0.019 0.019s 0.1113 0.0893 0.1088

kav (s-1) 5.8010-6 1.1510-5 1.0610-5

MSPDmodel

R2 0.9928 0.9982 0.9943

Spherical 0.1140 0.0911 0.1116

Cylindrical 0.1137 0.0907 0.1108sLinear 0.1123 0.0911 0.1100

Spherical 6.69 4.66 4.93

Cylindrical 4.81 3.37 3.56X (cm)

Linear 2.89 2.04 2.14

Spherical 0.9963 0.9996 0.9976

Cylindrical 0.9953 0.9994 0.9969

PSPD

model -uniform

length

R2Linear 0.9944 0.9990 0.9958

Spherical 0.1225 0.0934 0.1179

Cylindrical 0.1235 0.0933 0.1183sLinear 0.1263 0.0932 0.1127

Spherical 14.80 6.59 9.75

Cylindrical 11.07 5.05 7.38Xmax (cm)

Linear 7.09 3.35 3.83

Spherical 0.780 1.973 0.953

Cylindrical 0.699 1.643 0.843m

Linear 0.579 1.261 1.024Spherical 6.49 4.37 4.76

Cylindrical 4.55 3.14 3.38maxX1m

mX

+=

Linear 2.60 1.88 1.94

Spherical 0.9999 0.9999 1.0000

Cylindrical 0.9999 0.9999 1.0000

PSPDmodel -

variable

length

R2

Linear 0.9999 0.9999 0.9991

Table 3. MSPD and PSPD estimated parameters and goodness-of-fit for the three tall

column experiments.


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Comparison Factor Factor s Trend X (cm) Trend

9.5 mm 0.1712 2.030NA

19.1 mm 0.1132 3.364

9.5 mm 0.1305 3.387

1

A19.1 mm 0.1154

) 0 1 2 1 3

4 s

4.651

A 5 6 7 8 9 @ A @ B

8

7

9.5 mm 0.1179 3.254Without binder

19.1 mm 0.1150 4.968

9.5 mm 0.1431 3.5192

With binder19.1 mm 0.1168

8 9 @ A @ B

C sNo binder

B

C s3.385

No effect

Slow 0.1440 3.284NA

Fast 0.1405 2.109

Slow 0.1220 4.4343

AFast 0.1223

AB

C s

3.547

AB

8

7 8 D E

F G

9.5 mm 0.1324 3.461Slow 19.1 mm 0.1203 4.648

9.5 mm 0.1480 2.5964

Fast19.1 mm 0.1090

H I P Q P E

F s

3.733

H I P Q P E

H

G H D E

F G

Slow 0.1226 5.044Short bed

Fast 0.1173 3.303

Slow 0.1140 6.6865

Tall bedFast 0.1014

No effect

4.792

H R S

T U T V

S

W U

Slow 0.1174 4.844Without binder

Fast 0.1140 3.770

Slow 0.1300 3.7166

With binder Fast 0.1430

No binderS

W s

2.991

BinderS

W U

T V

S

W U

Table 4. Comparison of the means of six two-factor sets.

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