~ Pergamon0008-4433(95)00020-8

Canadian Metallurgical Quarterly, Vol. 34, No.4, pp. 303-310,1995Copyright 1995 Canadian Institute of Mining and Metallurgy

Printed in Great Britain. All rights reserved0008-4433/95 $9.50+0.00

A REACTOR MODEL FOR GOLD ELUTION FROMACTIVATED CARBON WITH CAUSTIC CYANIDE SOLUTION

T. M. SUNt and W. T. YEN~t Asia Minerals Corp., Vancouver, B.C., Canada

t Department of Mining Engineering, Queen's University, Kingston, Ontario, Canada

(Received 16 December 1992; in revised/arm 3 May 1994)

Abstract-A model for the Zadra elution process was developed based on pore diffusion inside the carbonparticles, the dispersed plug flow in the elution column, and the gold desorption from carbon as a firstorder reaction. A set of process description equations was obtained. Numerical solutions for the systemequations are given using the finite difference method, and the stability for the method was also analyzed.Fitting of experimental elution concentration profiles using this model was presented. The model can fitboth the concentration profile and the recovery curve, and can simulate the elution process up to the finalstage. It was found that the elution process is controlled by a first order rate constant and the effectiveintraparticle diffusivity in the carbon particle. The parameter analysis showed that the rate constant wasdependent on the elution temperature and the chemical composition of the eluant. The effective intraparticlediffusivity was relatively constant under the experimental conditions.

NOMENCLATURE NPeQ

BN,PN number of the equal intervals for column and particle, Qorespectively e,

C concentration of gold cyanide in the column solution r(mol dm :") rAu

C(t,y) concentration of gold cyanide in the column solution att, y (mol dm -3) R

Ccal calculated concentration of gold cyanide at the exit of the Recolumn (mol dm :") S

Cexp experimental concentration of gold cyanide at the exit of tthe column (mol dm -3) u

Cp concentration of gold cyanide in the carbon particle pore W(mol dm ") Wt

Cp(t,y, r) pore concentration of gold cyanide at t,y, r (mol dm ") ydp particle diameter (em)D I axial dispersion coefficient (em" S-I) IX,{3, WDe effective intraparticle diffusivity (cnr' S-I) (JD1B molecular diffusivity of gold cyanide in dilute solution r

(em? S-I) 8pk; rate constant for gold desorption (S-1 or h- 1) 8km external mass transfer coefficient in particle surface f1(em S-1) PL length of the carbon packing area in the column (em) Pcn number of carbon particles >

number of moles of gold in a subvolume at time tPeclet numbercarbon loading of gold (kg t- 1)initial carbon loading of gold (kg r ')average initial carbon loading of gold (kg r ')radial distance in carbon particle (em)rate of gold desorption per unit volume carbon (mol S-1cm- 3)radius of carbon particle (em)Reynolds numbercolumn cross-sectional area (em")time (s)interstitial flow velocity (em S-I)mass flux of gold (mol s-I cm -2)weight of carbon (g)distance along the column axial direction (em)

constantsconstriction factor for carbon porestortuosity of carbon poresporosity of carbon particlevoid fraction of the packed columnviscosity of solution (g cm- I S-I)density of solution (g em -3)density of carbon particles (g em:")sum of the squares of the deviations (g em -3)

INTRODUCTION commonly used methods in the gold industry are the Zadra,Anglo American Research Laboratories (AARL) and Micro

The carbon-in-pulp (CIP) process for the recovery of gold from processes. The Zadra process is carried out by pumping a hotcyanide leach pulps is used widely in modern gold-producing caustic cyanide solution through the column and then to anplants. The success of the CIP process is attributed to the electrowinning cell for gold deposition [1]. The barren elec-development of elution methods for gold and silver from the trolyte is recycled to the column as eluant. In the AARL elutionloaded carbon. Carbon elution is usually carried out at high process, the loaded carbon is acid washed and soaked in a hottemperature in a column in which the eluant is pumped through caustic cyanide solution for 1 h [2]. Hot de-ionized water is thena packed carbon bed in the upflow direction. The operation passed through the column. In the Micro process, the eluantis isothermally carried out in a batch mode. The three most contains large amounts of organic solvents such as methanol,

303

304 T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

acetone, or acetonitrile [3]. The elution requires 6-48 h forcompletion depending on the method employed. The long elu-tion time is primarily due to the slow diffusion of the auro-cyanide ion within the micropores of the relatively largeparticles of activated carbon.

Unlike the adsorption process, which has received con-siderable attention, little work has been done on the modellingof gold cyanide elution from activated carbon. The first attemptto model the gold elution in the Zadra process was presentedby Adams and Nicol [4]. They used a semi-empirical kineticmodel to quantitatively describe the elution kinetics, in whichthe pore diffusion in the carbon particles was not considered.The experiments were conducted in a batch stirred tank system,not in an elution column. A similar model was further studiedby Adams [5], as well as Vegter and Sandenbergh [6]. The effectsof chemicals on the Zadra elution were considered, but theelution profile was still not simulated very well. Some modellingwork has been performed on the AARL elution process [7,8]. A preliminary model for the AARL elution process wasdeveloped on the assumption that the elution behavior of goldwas dominated by the changing sodium concentration duringthe washing cycle [7]. Van der Merwe and Van Deventer [8]also demonstrated an intraparticle-film diffusion model with ashifting equilibrium, which could be used to simulate AARLelution curves.

This paper aims to analyze and establish a kinetic model forthe Zadra elution system. The elution kinetics are based on thepore diffusion model and the dispersed plug flow model. Afterthe development of a set of system description equations andits numerical solution, the model was affirmed by experimentalelution data.

KINETIC MODEL FOR CARBON ELUTION

The Zadra elution system is schematically shown in Fig. 1.The elution process can be divided into two parts: the plug flow

Loaded Carbon

Elution

Column

Heat

Exchanger

Eluted

Carbon

Fig. 1. The Zadra elution process.

of eluant in the packed column and the intraparticle diffusionflow of gold cyanide within the porous carbon particles to thesurface.

Dispersed plug flo\\' model

The eluant flow in the gold elution column is principally of theplug flow type superimposed with some degree of back mixing orintermixing because of the packed carbon particles. The flowfield may be modelled by a dispersed plug flow model. In theplug flow reactor, the gold is continually desorbed from thecarbon particles as eluant flows through the length of the reac-tor. The plug flow reactor may be divided into a number ofsubvolumes, and the reaction rate is assumed as spatially uni-form within each subvolume (as shown in Fig. 2). A subvolumelocated a distance )' from the entrance of the plug reactorcontains three types of material changes: bulk flow (in and out);dispersion (in and out) and desorption of gold from carbonparticles. A mass balance on gold cyanide in a subvolume atany instant in time t yields the following equation:

(rate of\ (rate of\in-out )bulk flow+ in-out )axialdispersion

(1)

(rate of gold desorptiOn) ( rate of )

+ from particles = accumulation .

The mass flow of gold cyanide per unit time may be defined asfollows:

entering by bulk flow = eliSCy;

leaving by bulk flow = ellSCy ...!'.y;

entering by axial dispersion = -(EDIS~~)I ;C) .I'

leaving by axial dispersion = -(eDIS~~)1 ;CJ y+!'.y

rate of desorption from carbon = (l-e)S~y(rAu);ac

rate of accumulation of gold cyanide = eS~Yat'

Substituting all these terms into eqn (1) and dividing by eS~ygives

C,H,-C, ~~I)HY-~~I,l-e ac-u .. .+D1 + --rAu = ---;::-.(2)

~)' ~)' E ct

~F(L),L----------:"

T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

Fig. 3. Single carbon particle.

As ~y .~ 0, the above equation becomes:

oC 02C oC 1-at = D1 oy2 -u oy + -G-rAu'

Pore diffusion model for gold desorption from carbon particle

The desorption of gold cyanide from a porous carbon particleinvolves the simultaneous processes of carbon desorption andpore diffusion. Assuming the carbon particle is spherical andthe radius is R, as shown in Fig. 3, the gold cyanide diffuses atlocation r through a small distance I1r per unit time. Forsimplicity, the desorption rate of gold cyanide at layer r insidea carbon particle can be assumed as a first order reaction withrespect to the carbon loading at any instant in time t:

where kc is the rate constant (S-l) and -oQ/ot is the golddesorption rate per unit weight of carbon.

Then, a mass balance (molls) for gold cyanide between randr+~r in the spherical particle is

( oCp 2)1 ( oCp 2)1-Dey4nr GP. - -Dey4nr Gpr r r r+~r2 oQ 2 oCp

-4nr ~rpCat = 4nr ~rGp7it. (5)

When ~r ~ 0, and rearranging the equation, we obtain thefollowing partial differential equation:

oCp =~ ~(r2De OCp)_ ~ oQ. (6)ot r2 or or Gp ot

The first term on the right hand side of eqn (6) representsthe normal diffusion and the second term is the rate of golddesorption. If the effective intraparticle diffusivity De is inde-pendent of the radial distance r, eqn (6) may be rearranged as:

oCp = De(02Cp +~OCp) _ ~ oQ .ot or2 r or Gp ot

When Cp, the gold concentration inside carbon particle pores,is found by solving eqn (7), the rate of gold desorption fromthe carbon particle in eqn (3) can be calculated from the averagemass flux of gold cyanide from the surface of the carbon particleto the bulk fluid in the column:

305

(8)

where W is the mass flux of gold cyanide (mol S-l cm-2) andkm is the mass transfer coefficient (cm S-l). Defining rAuas thegold desorption rate per unit volume of carbon in mol S-l cmc-3,i.e. rAu= W(4nR2)/(~nR3), we have

3kmrAu=:: -'R [Cplr=R - C]. (9)

Initialand boundary conditions

Before the loaded column is subjected to elution with a pureeluant, there is no gold in solution, either.in the pores of carhonor in the column. Thus, the initial gold concentration in theelution column is

C(t, y) = 0 at t = 0 ; (10)

the initial gold concentration in the pores of the carbon particleis

(3)Cp(t,y,r) = 0 at t = 0

and the gold loading on the carbon at the initial time is

(11)

Q(t, y, r) = Qo(r) at t = 0, (12)

(4)

where Qo(r) is the initial gold distribution in the carbon particlesalong the direction of the radius (r).

The gold elution column is a closed reactor in the Zadraprocess. Because the eluant is pumped into the column at a slowflow rate of two bed volumes per hour, there is no dispersion orradial variation in the eluant concentration, either upstream(closed) or downstream (closed) of the reaction section of thecolumn; The closed-closed vessel boundary conditions [9] aretaken for eqn (3) as follows:

at the outlet of the column,

oC-= 0 at y =L;oy

(13)

and at the inlet of the column,

D10CC = - - - at y = O.u oy

Boundary conditions (13) and (14) are the Danckwerts boun-dary conditions [10]. The spherical symmetry condition isapplied to the carbon particles:

(14)

oCp-= 0 at r =0or . (15)

At the surface of the carbon particle, the molar flux must becontinuous at the surface, thus:

(16)

(7)

PARAMETER ESTIMATION

Gold distribution within activated carbon particles

During the adsorption process, the gold cyanide penetratesthe carbon particle with a reasonably sharp front moving intothe particle with increasing time ..The gold distribution within

306 T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

the carbon particles is generally not homogeneous. It requiresa prolonged adsorption time to achieve a uniform distribution.According to the SEM analysis with a dispersive X-ray system[11], the distribution of gold in the carbon particle can beassumed as a linear distribution along the radial direction of aparticle at an adsorption time less than 24 h. Thus, the initialdistribution of gold on carbon, Qo, can be expressed as:

where Qav is the average initial carbon loading of gold, and Ris the average radius of the carbon particles.

Dispersion coefficient (D1)

In a packed column, such as in the Zadra elution process, thedispersion coefficient can be related to the Peclet number by thefollowing equation [9]:

udD1=-p'

8 e

where d is the particle diameter and Pe is the Peclet number.The following equation is a generally accepted correlation

for the axial dispersion in packed columns [12]:

0.2 0.011 048Pe=-+--Re' .8 8

Although the applicability of eqn (19) for a packed columnunder desorption conditions was not verified, this equationsuggests that the Peclet number for axial dispersion under thepresent condition is approximately independent of the flowvelocity under elution conditions, owing to the very low particleReynolds number (Re) for the elution system.

External mass transfer coefficient

The external mass transfer coefficient km in eqn (9) can beestimated by the following correlation [13]:

(DO )0.67

km = 1.09 8;pB UO.33,

where D iB is the molecular diffusivity of gold cyanide,Au(CN)2' in dilute solution, 1.8x 10-5 cm2 S-I at 61C [14]. Itcan be converted to higher temperatures by eqn (21) [9]:

where /1b and /12' are the liquid viscosities at temperatures T1and T2, respectively.

NUMERICAL SOLUTIONS FOR THE PARTIALDIFFERENTIAL EQUATIONS

Partial derivatives can be approximated by finite differencesin many ways. All the approximations introduce truncationerrors. Equations (3), (4) and (7) can be reduced to "equivalent"discrete equations using uniform space increment ~y in the y

(17)

direction, 111' in the radius direction and uniform time increment~t in the t direction.According to Carnahan et al. [15],the derivatives with respect

to t, y and l' are replaced in a particular way by finite differenceratios, dividing the range of R into PN equal intervals of ~r,and dividing the range of L into BN equal intervals of ~y.Cp .. is the gold concentration at column position j and insidepa~iicle position k at time i. The values of j and k are equal too at t = 0 and increase by 1 as the time increment passes by !1t.Ci.j is the gold concentration at the column positionj at time t.For 1 ~j ~ BN and 1 ~ k ~ PN -1, eqn (4) becomes:

(22)

For eqn (7):

k-l k+lC .. =w--C .. +(1-2w)C .. +w--C

P/.1ok k P\I-l).j.(k-l) P\I-I).j); k PI/-I}.).,'.()

(18)where

I1tw=D--

e (111')2 .(24)

(19)Then, for eqn (3) we have

Ci,j = (rt.-!3)Cu-1),u+I)+(1-2rt.)Cu-I),j

1-8+ (rt.+ !3)CU-I) U-I)+--l1t 1'Au, (25). 8

where

I1trt.=D1--, (26)(11.1')2

~t13 = II 211y' (27)

And for eqn (9) we have

(20) (28)

(21)

For the interface point, Cp .. was derived as shown in theappendix. l.j.P:-;Because explicit representations have been used for the

numerical solutions, the results computed by eqns (23) and (25)must be examined for stability. For the dispersed plug flowmodel, mass conservation dictates that the first three coefficientson the right side of eqn (25) sum to one for all values of DI~ ~tand l1y.According to the method detailed previously [16, l7]~it was found that (rt.-!3), (1-2L1.)and (LI.+!3)should be positiveand (rt.-!3)+(1-2L1.)+(rt.+!3) ~ 1 to avoid instability. Hence,the stability criteria are:

!1t 1D --~-

1 (~y)2 ~ 2'(29)

D1 II-~-l1y ~ 2' (30)

Because the presence of a lower-order linear term in the para-

Table 1. Specifications of the loaded carbon samples

T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON 307

Sample A Sample B Sample C

Qo (kg t-I)BpB

rc (g cm-3)

8.030.550.40.45

9.630.550.40.45

4.610.550.40.45

bolic partial differential equation does not influence the stabilitybounds [18], the stability constraint for equation (23) is givenas

~t ID--~-e (~r)2 " 2'

The truncation error tends to zero as ~r, ~y and ~t go tozero. This implies that the solutions of the finite differenceequations converge to the exact solution of the partial differ-ential equations as ~r, ~y and ~t go to zero. It is also notedthat the requirement of convergences places a severe restrictionon the interval sizes in the rand y direction and hence resultsin long computing times.

FITTING TO EXPERIMENTAL RESULTS

Fitting the numerical results to the experimental results isconducted using the least squares method. The objective of thismethod is to minimize the sum of the squares of the deviationsbetween the experimental and theoretical elution profiles. Thisis expressed mathematically as

n

= I [Cexp - Ccalf,k=1

where Cexp is the experimental result and Cca1 is the gold con-centration predicted by the model at the exit of the elutioncolumn.To minimize the objective function for the given search

parameters, the combination of two series of search values, Deand kc' are calculated on an IBM mainframe with a FORTRANprogram.Experimental data from a laboratory elution column (2.5 cm

dia., 20 cm high) were used to examine the model. A mass of50 g of loaded carbon was packed in the column, which wasthen filled with water and heated to the desired temperaturewith an electrical heating pad. The temperature was maintainedat 2C with a Parr temperature controller. The eluant wasprepared by dissolving analytical grade reagents in de-ionizedwater. The elution flow rate of two bed volumes per hour wascontrolled by compressed nitrogen gas.Three different loaded carbon samples were obtained from

the production CIP circuits of Canadian gold mines. The speci-fication of the carbons is shown in Table 1. Other parametersused in the model, such as particle porosity, void fraction ofpacked column and density were assumed to be the same forall samples at no measurements.The average equivalent particle radii, R, were determined by

measuring the weight (Wt) of about 200 randomly selectedparticles (n) and calculated by the following equation:

(33)

Equations (3), (4) and (7) were used to calculate the goldconcentration profile and recovery as a function of time. Figures4(a), 5(a) and 6(a) demonstrate the gold concentration profileof the eluant at the exit of the column. The correspondingrecovery curves for the elution process are shown in Figs 4(b),5(b) and 6(b). The results demonstrate that the calculated curvesagree well with the experimental data, both in concentrationprofiles and recovery curves. The recovery curve is the inte-grated desorption curve of the gold concentration against thebed volume of the elution. It is apparent that both De and kc

(31)

(32)

1500

1350....J

o experimental........0'1 1200E - fitted curvec

1050.2"

k =0.70 h-1

C 900 c -5 2Q) De=1.11x10 em /suc

7500u::J 0 600Q)

a::J 450W

300 0

150

00 2 3 4 5 6 7 8 9

Time (Hour)

Fig. 4(a). Elution of gold from the carbon with NaCN-NaOH :NaCN,1 g 1-1; NaOH, 10 g 1-1; 148C; Q = 9.63 kg t-1 Au.

100

90

80~ o experimental-0.

70 - Calculated0() -1'0 60 k=0.70 hc -5 2~ De=1.11x10 em /s(])

50>0u 0(])~ 40

30

20

10

2 3 54 6 7 8 9

Time (Hour)

Fig. 4(b). Elution of gold from the carbon with NaCN-NaOH: NaCN,1 g 1-1; NaOH, 10 g 1-1; 148C; Q = 9.63 kg t-I Au.

T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

345

Time (Hour)

Fig. 5(a). Elution of gold from the carbon with NaCN-NaOH: NaCN,4 g 1-1; NaOH, 10 g 1-1; 148C; 2 BV h-I.

345 6

Time (Hour)

Fig. 5(b). Elution of gold from the carbon with NaCN-NaOH: NaCN,4 g 1-1; NaOH, 10 g 1-1, 148C; 2 BV h-1

308

1500

1350....J

"-01 1200Ec-

1050.2"C 900Q)uc

7500u:J

0uQ)

400:::

30 0

20

010

00 2 3 4 5 6 7 8 9

Time (Hour)

Fig. 6(b). Elution of gold from the carbon with NaCN-NaOH: NaCN,2 g 1-1; NaOH, 10 g 1-1; 140::C; 2 BV h-I.

system [eqns (3), (4) and (7)] is accurate for the gold elutioncolumn.

A long tail in the elution profile, as shown in Figs 4(a), 5(a)and 6(a), indicates that the elution of a small portion of thegold cyanide is delayed by the pores inside the core of thecarbon particles. The demonstrated ability to fit the long tail isseen as an indication of the importance of the porous diffusionin the elution of gold cyanide from loaded carbon. The sug-gested model can simulate the complete elution process. Othercarbon elution models, such as that seen in Ref. [4], are hardlyapplicable to the final stage of elution, owing to neglect of theporous diffusion inside carbon particles.

T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

In this model, the elution profile is characterized by the effec-tive intraparticle diffusivity, De' and the rate constant for golddesorption, kc. Using the fitted parameters it is possible toquantitatively study the effect of experimental conditions onthe elution process, which can provide a better understandingof the mechanism of gold cyanide elution from activatedcarbon. The experimental conditions and the fitted parametersare summarized in Table 2.

From Table 2 it is clear that, although the initial carbonloading and the chemical conditions were different, the effectiveintraparticle diffusivities were the same within the range ofexperimental conditions and samples tested. Hence, the rateconstant, ko of gold desorption in the carbon is the key factorfor the gold elution, and it is a function of the elution tem-perature and chemical composition of the eluant.

The effective diffusivity value of 1.11 x 10-5 cm2 S-1, whichwas caculated from the experiments, is smaller than the goldcyanide diffusivity of 5.06 x 10-5 cm2 S-1 estimated by eqn (21).This is because the pores in the carbon particles are not straightand cylindrical; rather, they consist of a series of tortuous,interconnecting paths of varying cross-sectional areas [19]. Theeffective intraparticle diffusivity in the study can be related tothe physical properties of carbon by eqn (34) [9]:

D1B(JD =--e ,

!

where (J is the constriction factor, which accounts for the vari-ation in the cross-sectional area of carbon particle pores and isa function of the ratio of maximum to minimum pore areas,and! is the tortuosity.

Typical values of the system are: ! = 3-5, (J = 0.8 [9], soDe/D1B = 0.26-0.16. For the experiments, the ratio of De/D1Bis 0.22, which indicates that the pores inside the carbon particlesreduce the diffusivity of gold cyanide complex within the carbonparticles to about 22 % of that in a normal dilute solution. Thisfurther confirms the importance of pore diffusion within thecarbon particles for the gold elution process.

CONCLUSIONS

A model for the Zadra carbon elution process was developedand the system description equations and the initial and boun-dary conditions have been defined. In this model, the gold

Table 2. The experimental conditions and the fitted parameters

Sample A Sample B Sample C

Initial carbon loading:Qo (kg t-I) 9.63 8.03 4.61

Elution conditions:Temperature caC) 148 148 140NaCN (g I-I) 1.0 4.0 2.0NaOH (g I-I) 10 10 10Flow rate (BV h -1)* 2.0 2.0 2.0

Fitted parameters:kc (h-I) 0.70 0.78 0.40Dex 105(cm2S-I) 1.11 1.11 1.11

* BV h-1, bed volume per hour.

309

desorption process, the porous diffusion and the dispersed plugflow in a gold elution column have been considered. The golddesorption was assumed to be a first order reaction.

A numerical solution for the model using the finite differencemethod has been obtained. The stability conditions were alsogiven. The model has been programmed in FORTRAN on anIBM mainframe.

The elution profiles of experimental and calculated resultshave a reasonable fit. The model can describe the com-plete elution process, including the long concentration tailin the elution profile at the final elution stage. The modelhas two parameters, De and kc' besides the physical andinitial parameters. The kinetics constant kc is related to theelution conditions in the column. The effective intraparticlediffusivity of gold cyanide in carbon is not significantly affectedby the composition of the eluant with these experimentalconditions.

Acknowledgements-The research was partially supported by the Natu-ral Science and Engineering Research Council of Canada (NSERC).The authors wish to thank Dr R. M. Erdahl in the Department ofMathematics and Statistics in Queen's University for many helpfuldiscussions.

(34)

REFERENCES

1. J. B. Zadra, A. L. Engel and H. J. Heinen, Rep. Invest. Bur. Mines.4843 (1952).

2. R. J. Davidson and D. Duncanson, J. S. Afr. Inst. Min. Metall. 77,254 (1977).

3. D. M. Muir, Hydrometallurgy 14(1), 47 (1985).4. M. D. Adams and M. J. Nicol, Gold 100 Proc. Int. Con! on

Gold, Vol. 2, Extractive Metallurgy of Gold. SAIMM, p. 111,1986.

5. M. D. Adams, Trans. Inst. Min. Metall., Sect. C. 99, C71 (1990).6. N. M. Vegter and R. F. Sandenbergh, Hydrometallurgy 28(2), 205

(1992).7. W. Stange, Miner. Engng 4(12), 1279 (1991).8. J. S. J. Van Deventer and V. E. Ross, Miner. Engng 4(7-11), 667

(1991).9. H. Scott Fogler, Elements of Chemical Reaction Engineering, 2nd

Edn. Prentice-Hall, pp. 543-92, 759-94, 1992.10. P. V. Danckerts, Chem. Engng Sci. 2, 1 (1953).11. N. M. Vegter, Hydrometallurgy 30, 229 (1992).12. C. Y. Wen and L. T. Fan, Modelsfor Flow Systems and Chemical

Reactors. Marcel Dekker, p. 570, 1975.13. E. J. Wilson and C. J. Geankoplis, Ind. Engng Chem. Fundam. 1,9

(1966).14. G. M. Schmid and M. E. Curley-Fiorino, Encyclopedia of Electro-

chemistry of the Elements, Vol. IV. Marcel Dekker, pp. 87-178,1975.

15. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied NumericalMethods. John Wiley & Sons, NY, pp. 429-519, 1969.

16. A. Constantinides, Applied Numerical Methods with Personal Com-puters. McGraw-Hill, pp. 458-519, 1987.

17. B. A. Finlayson, Nonlinear Analysis in Chemical Engineering.McGraw-Hill, p. 216, 1980.

18. L. Lapidus and G. F. Pinder, Numerical Solution of Partial Dif-ferential Equations in Science and Engineering. John Wiley & Sons,NY, pp. 149-350, 1982.

19. G. J. McDougall and C. A. Fleming, in Ion Exchange and SorptionProcesses in Hydrometallurgy (edited by M. Streat and D. Naden).The Society of Chemical Industry, pp. 56-126, 1987.

310

APPENDIX

T. M. SUN and W. T. YEN: GOLD ELUTION FROM ACTIVATED CARBON

Interface

Finite-difference approximations at the interface between carbon andsolution

When the gold cyanide diffuses from the core of the carbon particleto its surface, the mass flow at the interface between carbon and solutionbehaves as shown in Fig. AI. We wish to derive the relevant finite-difference approximation for gold concentration Cp at point PN on theinterface between carbon and solution. The following procedure isbased on the continuity of molar flux at the interface.

In carbon (from Taylor's expansion) we have, approximately,

C ~ C _ dr(aCp) + (dr)2 (a2Cp) Pi,j.(P:-i-I) Pi.j.P:-i ar 2 '" 2 (AI)

PN.c or PN,c

The subscript (PN, c) denotes the derivative in carbon at the interface,i.e. :

(a2C) 2 [ (ac) ]-_P ~-- C -C +dr _P .a 2 (A)2 Pi.j.(P:-i-ll Pi,j.P:-; arl' PN.c IJ.r PN,cAlso, the time derivative is approximated by

(acp) ~ Cp(i+ll.j.P:-i - CPi.j.P:-;at PN,c - dt .

Substitution of eqns (A2) and (A3) into eqn (4) gives

_C_P(i+_!l-'----.j.P_:-i-_Cp-'----ij_.P:-;= D(_2_[c -C +.1r(_aC_p)]dt e (dr)2 P(i+l).j.(P:-;-I) P(i+I).j.P:-; ar PN.c

Carbon solution

PN-2 PN-l PN

Fig. A I. Interface between carbon and solution.

+ _2_ (8:p) ) Pc Qu ...I),j.PN - Qi,j.PN (A4)PN . .11' cr PN,c l;p dt

(A2)Then, from eqns (A4) and (7), it follows that

C = [rc. +?wC - ~(Q('. I) .p .-Q. 'PN)]I(1 +C+2w)PU.I),j.P:-; ~ I,J - P(i-I).j.(P:-;-1l p 1- ,J,.~ t,}, - ,

(AS)

(A3) where

v _ 2(1 +PN)dt k~ - PN.1r m

Dedtw=--.

(dr):!

(A6)

(A7)