hydrometallurgy
ELSEVIER

Hydrometallurgy 36 (1994) 285-294

A general shrinking-particle model for the chemical
dissolution of all types of cylinders and discs
C. N f f i e z a'*, M. Cruells b' 1, L.

Garcia-Soto a

aE. T.S. de Ingenieros Industriales, Mendizabal s/n (Esteiro), 15403 Ferrol, La Coruga, Spain
bFacultad de Quimica, Departemento de Ingenieria Quimica y Metalurgia, Avda. Diagonal 64 7,
7° Piso, 08028 Barcelona, Spain

Received October 12, 1993; revised version accepted January 14, 1994

Abstract
There is a manifest lack of models with exact solutions for describing the kinetic behaviour of cylindrical solids in reactions under chemical control and for determining the kinetic constants of the reaction system. This paper deduces two models that offer exact
solutions: one for any type of cylinder with a diameter less than the height and another for
any type of disc with a diameter greater than the height. It is demonstrated that the approximate kinetic model proposed in the literature for cylinders is actually the exact model
for a regular toroid. Finally, we propose a single model for any type of cylinder or disc,
making use of the so-called kinetic sphere radius.

I. Introduction
Deducing simple kinetic models for the chemical dissolution of bodies of various geometric shapes is one of the most primary and elementary of tasks in heterogeneous kinetics and has become part of the body of doctrine in the field. In
spite of the large number of models proposed, very little attention has been paid
to the deduction of a kinetic model for dissolving cylinders. At present we have
no exact model for the dissolution of cylinders or discs and only approximate
solutions are available [ 1-3 ].
For the purpose of modelling the behaviour of a cylindrical solid when it is
dissolved in a reagent under chemical control, it has been assumed that, for a
cylinder of sufficient length, such as a wire, the input of the bases to the amount
tAuthor for correspondence.
0304-386X/94/$07.00 © 1994 Elsevier Science B.V. All;  rights reserved
SSDI 0304-386X ( 94 )00006-0

C. Nft~ez et aL / Hydrometallurgy 36 (1994) 285-294

286

[_

"1%

Io

j

"

Fig. 1. The dimensions of interest in modelling a cylinder of/o> 2ro.

of solid dissolved is negligible, which allows the solid to dissolve only according
to the area of the cylindrical surface.
This means that the length of the cylinder (l) can be taken as constant and only
the radius of the cylinder base (r) diminishes during the reaction. Under these
conditions, the following model can be deduced for a reaction like the one described in Eq. (2) below:
1 - (1 - x )

½=flkqC~'t
paro

(1)

where: x = fraction reacted; fl= stoichiometric coefficient; kq = chemical constant
(varies only with temperature); C~, = activity of the reagent in the fluid phase,
raised to its order, a; PB = molar density of cylindrical solid; t = time; ro and lo are
the initial radius of the base and the initial length of the cylinder, respectively
(Fig. 1).
As shown below we will see that this approximate model for sufficiently long
cylinders, as presented in the literature, is actually the exact kinetic model for the
dissolution of a toroid under chemical control.
The model in Eq. (1) begins to show considerable errors when the cylinder
length approaches the value of its diameter. It becomes unacceptable--not because of larger errors but because it is not applicable--when the cylinder has a
size of 2ro > 10. Those cases in which the cylinder diameter is greater than its height
are perhaps the most interesting for industry and, as yet, are lacking an adequate
kinetic model as simple as other accepted models currently in use.
This paper presents two models: one for cylinders and another for discs, but
they are the same and are exact and valid for any cylinder or disc.

2. A general kinetic model for cylinders
This model will be applied, in principle, only to cylinders of diameter 2ro and
height lo, such that: 2ro < lo.
Let us consider a solid (B), compact, kinetically isotropic and homogeneous,
which is dissolved under chemical control in a reagent (A), present in the fluid
phase to give certain products, according to the following stoichiometry:

C. N~fiez et al. /Hydrometallurgy 36 (1994) 285-294
Ali q -~- flBso I ~-. Products

287

(2 )

From a batch mass balance, let v be the reaction rate, defined by:
V=

1 dN A k C '~

S

~

-~-- q

A

(3)

where: S = t h e solid surface area and NA= number of moles of reagent A; other
symbols as in Eq. ( 1 ).
Since:
fldNA =dNB

(4)

dNs =pBdV

(5)

where: NB = number of moles of reagent B; V= volume of solid B. From this point
on, any magnitude bearing the subscript o refers to its value at the initial moment,
t=0.
Combining Eqs. (4) and ( 5 ) with Eq. ( 3 ) and solving, we get [ 4 ]:
v

f dV

y=L-[flkqC~']
-jt=v,t

(6)

Vo

To integrate the first member of this equation, let us assume a cylinder like the
one in Fig. l, where h is the uniform thickness reduced by a certain time of chemical reaction [ 5 ], so that:

r=ro - h

(7)

l=lo-2h

(8)

With these equations, the value of l in terms of r will be:

l=(lo-2ro)+2r

(9)

The expressions for the volume and surface area of the cylinder, respectively,
are:

V=rcr2l

(10)

S= 2~rl + 2rcr2

( 11 )

Substituting the value of l, as given by Eq. (9), in these two equations we get:
V= ~(10 -2ro)r2+2~r 3

(12)

S= 2~( lo - 2ro )r + 6gr 2

(13)

Differentiating Eq. ( 12 ) with respect to r and dividing it by Eq. ( 13 ), we get:

d--~=dr=-dh

(14)

C. Nt~gezet aL / Hydrometallurgy36 (1994)285-294

288

Substituting this value of d V/S in Eq. (6) and solving, the result is:

l-r=FflkqC~q~-Vlt
ro l PB Aro-ro

(15)

This is the valid kinetic model for the dissolution of all /o>2ro cylinders.
Nevertheless, to follow up the reaction by chemical analysis, the relation between
the cylinder radius and the fraction reacted is as follows:

r21=ro2(1-x) lo

(16)

Substituting the value of I as given in Eq. (9), and ordering, we get:
r3k

( / o - 2 r o ) 2 ro2lo,,
2
r ----~-t,--x) =0

(17)

Eq. ( 17 ) enables us to calculate the value of r that satisfies Eq. ( 15 ) for each
experimental value of x. By obtaining a sufficient number of values for r we can
plot r o - r against t and, adjusting the points to a straight line, we can find the
slope (v 1 )--the brackets in Eq. ( 15 ).
We can get values for v~ by carrying out experiments at various values of CA.
Using the logarithm of v~ with these data, we get the following:
logv I

~"

log flkq + txlOgCA
PB

( 18 )

Experimental adjustment to this straight line enables us to find the chemical
constant (kq) and the order ( a ) of reagent A, and several values of kq obtained
in experiments at constant CA and different temperatures will allow us to determine the activation energy.
3. A general kinetic model for discs

Let us consider a disc like the one in Fig. 2, in which 2ro > lo. After a certain
time the chemical reaction makes the disc thinner; in this case the greatest input
to the dissolution comes from the solid dissolved at the base of the cylinder. The
disc finally disappears w h e n / = 0 , but at this time r > 0 and we cannot use the
model of Eq. (15 ) to describe the kinetic behaviour of the disc. Since the disc
begins to react when l = lo and disappears when l = 0, the model to be deduced
must be expressed in terms of/. To do this we solve for r in Eq. (9) and substitute
this value in Eqs. ( 10) and ( 1 1 ) to arrive at the following equations for the volume (V) and surface area (S) of the disc:

V=4[13-2(lo-2ro)/2+ (lo - 2ro)2l]

(19)

S=21312-4(lo - 2ro)/+ ( l o - 2 r o ) 2]

(20)

C. Nft~ez et al. / Hydrometallurgy 36 (1994) 285-294

I ~

289

,_ ro
i

21

m

/

h~

u~!l

L_J I

Fig. 2. The dimensions of interest in modelling a disc of 1o< 2ro. R is the radius of a sphere inscribed
in the disc, or the kinetic radius.

Differentiating with respect to l in Eq. ( 19 ) and dividing it by Eq. (20), the
result is:
dV dl
--dh
S
2

(21)

If we say that l/2=R, with R as the radius of a sphere inscribed in the disc
tangent to its two bases, called a kinetic sphere [4], and therefore a function of
the radius of the kinetic sphere, then dl/2 = d R and lo/2 =Ro; substituting the
value of d V/S in Eq. (6) with dR and solving, we get:
R
flkqC~] t v ~
I --~o = [ ~ _ ~ o
=~ool

(22)

This is the general kinetic model for all discs of the type 2Ro = lo < 2to in terms
of the radius of the so-called kinetic sphere (R), which takes a shape like the one
inEq. (15).
The chemical follow-up of the reaction is based on the ratio between volume
and fraction reacted (see Eq. 16 ), substituting l or lo with 2R or 2Ro, respectively,
and r with its value in (ro-Ro) +R. Substituting these values, operating and ordering, we get:
R 3 - 2 (ro -Ro)R2 4- (to -Ro)2R-roR~( 1 - x ) = 0

(23)

For each experimental value of x, Eq. (23 ) gives the corresponding value for
R, which satisfies Eq. (22). From here on we proceed in the same way as described for finding kq and o~ in the case of cylinders.
4. D i s c u s s i o n

A particular case in the family of cylinders is the one in which 2ro = 1o (an 'isometric' cylinder). In this case, Eqs. (15) and (22) are still valid, since ro=Ro

C. Nfi~ez et al. / Hydrometallurgy 36 (1994) 285-294

290

and r=R; that is, the radius of the sphere inscribed in the cylinder (R) coincides
at all times with the radius (r) of the cylinder base. Nonetheless, an exact model
has been proposed for this isometric cylinder [6 ] in terms of an initial equivalent
radius of the solid and its sphericity factor.
Note that ro in Eq. ( 15 ) and Ro in Eq. (22) are actually the radius of the sphere
inscribed in the cylinder and in the disc, respectively; therefore, these two equations are exactly the same, if we consider that the r or R that must always be
introduced into the model is the radius of the inscribed sphere.
For the purpose of comparing the approximate model ofEq. ( 1 ) with the model
for a toroid, we can briefly deduce the kinetic model for dissolving a toroid like
the one in Fig. 3 under chemical control. At any given time during the reaction,
the volume of the toroid is given by:
7~2

V=-~(a+b)(b-a) 2

(24)

and the surface area by:

S=nZ(a+b)(b-a)

(25)

Considering that for the toroid:

(a+b)=(ao +bo)
b-a=2r

(26)
(27)

Eqs. (24) and (25) become:

V=n2(ao +bo)r 2

(28)

%,\

•

I,

_

'

i/

Fig. 3. The dimensions of interest in the cross-section o f a toroid.

C. N~t~ez et al. / Hydrometallurgy 36 (1994) 285-294

S=2nZ(ao +bo)r

291

(29)

Differentiating Eq. (28) for r and dividing it by Eq. (29), we get:
dV=dr= -dh
S

(30)

Substituting this value for d V/S in Eq. (6) and integrating, we get Eq. ( 15 ). The
basis for expressing it in terms of fraction reacted is: V= Vo( 1 - x ) . From this we
deduce that:

r=ro(1-x) ½

(31)

If we then substitute this value in Eq. ( 15 ), we get Eq. ( 1 ), which turns out to be
the exact model to describe the reaction of a toroid under chemical control.
In order to look at the x curves (fraction reacted) against time (t), we chose a
range of values for ro and lo, but always with the condition that the volume of the
bodies analyzed is the same and equal to 10n:
( 1 ) a reference sphere: V= 10n; ro= 1.957;
(2) a cylinder of lo>2ro: V= 10n; ro= 1.581; •0=4;
(3) a disc oflo< 2ro: V= 10n; ro=2.236;/0=2;
(4) a cylinder like (2) but with the approximate Eq. ( 1 ) applied to it, using
the actual data for the fraction reacted of cylinder 2;
(5) a toroid: V= 10n; ro= 1; bo= 1 + ( 5 / n ) ; ao= ( 5 / n ) - 1.
We took the arbitrary value of vl =0.2 for all cases.
We then proceeded to calculate the fraction reacted (x) in terms of time (t),
and the values of vtt/ro in terms of the fraction reacted for each of the five bodies
listed above.
( 1 ) For the reference sphere we took:
V i i _ 1_ ( l _ x ) ~

(32)

r0
The final time that it takes for this sphere to dissolve is calculated from Eq.
(7), using r = 0; considering that h = v~t and solving for t, we get: t = 9.787.
(2) The final dissolution time for the cylinder will be: t = 7.906.
(3) The final dissolution time for the disc, depending on the dimensions chosen, will be: t = 5.
(4) The actual values for the fraction reacted of cylinder 2 were applied to the
approximate model in Eq. ( 1 ), calculating v~t/ro. In this case the dissolution time
for the cylinder is the same as for cylinder 2: t = 7.906.
(5) We used Eq. ( 1 ) for the toroid, which is its exact equation. The final dissolution time for the toroid is: t = 5.
Fig. 4 shows the fraction reacted (x) plotted against time. In the first place, the
final dissolution time depends on the value chosen for the minimum thickness of
each body. We can see that the sphere, as a body of minimum surface area per
unit of volume, is the one that takes the longest to dissolve, when it and all the

C. Nt~ftez et al. /Hydrometallurgy 36 (1994) 285-294

292

),----0-- O-

/

016
/
~
Or4
/

0
A
•
•

1
2
3
4

•

5

0¢2

t
I

I

2

I

I

4

i

I

6

I

I

I

8

Fig. 4. The fraction reacted (x) plotted against time (t) for the bodies referred to in the text. 1 = sphere;
2 = cylinder; 3 = disc; 4--- cylinder; 5 = toroid.

bodies are of the same volume. The x/t curves for discs and toroids give larger
fractions reacted for each reaction time since they have much more surface area
per unit of volume than the sphere. The fraction reacted for cylinder 4 calculated
with the value of x taken from the approximate model of Eq. ( 1 ) leads to errors
by default, with lower fraction reacted predictions than for cylinder 2 and even
for the sphere, which is unacceptable. The x/t curve for cylinder 2 is the correct
one and does not differ greatly from the curve for the sphere since we chose a near
equiaxial cylinder in order to see clearly the consequences of ignoring the reaction on the base area.
The straight lines obtained in Fig. 5 terminate at the final dissolution time values previously chosen for the bodies. We can see that the approximate model of
Eq. ( 1 ) applied to a cylinder with dimensions like number 2 does not give a
straight line. Nevertheless, up to fraction reacted values of 0.8 we can assume that
the slope corresponds to the dotted line, which would lead to errors of up to approximately 30% in estimating kq for the cylinder being analyzed.
Finally, since real solids are not uniform or rigorously isotropic, dissolution is
not homothetic and the reaction takes place preferably in areas that are more
active in energy terms, leading to pitting, caused by the presence of impurities,
concentration of dislocations, residual stresses, etc. Nevertheless, we can expect
that for low concentrations of defects and moderate fraction reacted rates, we can
use these models for an acceptable determination of the chemical constants in a
reaction. Furthermore, this is the first time that an exact model has been pre-

C. Nftgez et al. I Hydrometallurgy 36 (1994) 285-294

,,o,.,.

293

i, _-;'°/ /

//..//

o,8

/

/,*

/

d

IS//
OI4

Oa2

5

t
I

I

2

I

I

4

I

I

6

I

I

I

8

Fig. 5. Plot of vlt/ro against time (t) for the bodies referred to in the text. 1 =sphere; 2=cylinder;
3 = disc; 4 = cylinder; 5 = toroid.

sented to describe the kinetic behaviour of cylinders, discs and toroids of all types,
which completes the set of simple basic models in heterogeneous kinetics.

5. Conclusions
The mathematical treatment presented in this paper makes it possible to propose a single model, in terms of the radius of the inscribed sphere, to describe the
kinetic behaviour of cylinders and make exact predictions for the behaviour of
cylinders, discs or toroids in reactions under chemical control. It can also be used
to determine the kinetic constant, the order of reagent A and activation energy of
the reaction system, as long as the solid is reacted uniformly, because in the basic
models of heterogeneous kinetics the solid is assumed to be pure, homogeneous
and isotropic.

References
[ 1 ] Levenspiel, O., Ingenieria de las Reacciones Quimicas. Table 12-1. Revertr, Barcelona ( 1974 ),
p. 409.
[ 2] Sohn, H.Y. and Wadsworth, M.E., Rate Processes of Extractive Metallurgy. Plenum Press, New
York (1979), p. 144.
[ 3 ] Habashi, F., Principles of Extractive Metallurgy. Vol. I. Gordon and Breach, New York ( 1969 ),
p. 137.

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C. N ~ e z et al. / Hydrometallurgy 36 (1994) 285-294

[4] Ntifiez, C., Vifials, J., Roca, A. and Garcia-Soto, L., A general shrinking-particle model for the
chemical dissolution of crystalline forms. Hydrometallurgy 36 (1994), in press.
[ 5 ] B. Delmon, Introduction ~ la Cin6tique H6t6rogbne. Technip, Paris ( 1969 ), 281 pp.
[ 6 ] Ntifiez, C. and Espiell, F., The shape of bodies and its consequences on the chemical attack of
solids. Chem. Eng. Sci. 41 (8) ( 1986): 2075-2083.