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Unformatted text preview: terior of the particle, where a Fickian diffusion process can be
As suggested by the Goto, Roy, and Hirose (cited above), the interior diffusion process
can be viewed as one in which the rate of diffusion in the non-extracted interior core of the
particle is much slower than in the outer shell of the particle where most of the solute has been
extracted. This results in a sharp boundary or interface between the inner core and outer shell. In
the inner core, the concentration of solute is at its initial value. With time, the core region
shrinks as the sharp boundary progresses toward the center of the particle. This is the basis for
the so-called shrinking-core model widely used in modeling leaching operations. The model was
first conceived by Yagi and Kunii in 1955 ["Fifth Symposium (International) on Combustion,"
Reinhold Publishing Corp., NY (1955), pp. 231-244] for application to gas-solid combustion,
and extended to liquid-solid leaching by Roman, Benner, and Becker in 1974 [Trans. Soc.
Mining Engineering of AIME, 256, 247-256 (1974)]. In its general form, takes into account both Exercise 11.26 (continued) Analysis: (continued)
internal and external mass-transfer resistances. Let us develop the model for the case of
negligible mass-transfer resistance in the liquid external to the particle.
Assume that drc /dt, the rate of movement of the interface at the particle radius, rc , is
small with respect to the diffusion velocity of solute A, through the outer shell of the particle.
This is referred to as the pseudo-steady-state assumption. The importance of this assumption is
that it allows us to neglect the accumulation of solute as a function of time in the outer shell layer
as that layer increases in thickness, with the result that the model can be formulated as an
ordinary differential equation rather than as a partial differential equation. Thus, the rate of
diffusion of solute A through the outer shell is given by Fick's second law, (3-74), ignoring the
term on the left-hand side and replacing the molecular diffusivity with an effective diffusivity,
De, for the solute through the solvent in the complex matrix of natural material:
De d 2 dc A
r 2 dr
where cA is the concentration of solute in the outer shell of the particle and r is the radial distance
from the center of the particle. The boundary conditions are:
c A = c As = c Ab at r = rs c A = c A0 at r = rc
where the subcripts are s for the particle surface, b for the bulk, 0 for initial condition, and c for
the interface between the outer shell and core of the particle. These boundary conditions hold
because the mass-transfer resistance in the liquid film or boundary layer is assumed negligible
and the concentration of solute in the core remains at its initial value as the core shrinks.
If Eq. (1) is integrated twice and the boundary conditions are applied, the result after
simplification is: rc
c A = c A0 − c A0 − c Ab
To obtain a relationship between the location of th...
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