26 / 36
24
Chemical Technology • May 2016
MINERALS PROCESSING AND METALLURGY
low level. A typical purge bin and the design relationships
are shown in Figure 1 below.
Volatiles removal should be considered a ‘rate limited
process’. In addition to equilibrium considerations, time and
mass transfer coefficients are also important. Rate-limited
processes obey the fundamental relationship shown below:
R = K*DF
(1)
where: R = the rate of volatiles removal
K = a constant which depends on solids
morphology and type of drying equipment
DF = the driving force for removal of volatiles from
the solid. This is usually the difference between
the actual concentration and the equilibrium
concentration of the volatile in the solid.
A more specific form of the equation is shown below:
dX/dt = - k*a*(X-X
E
)
(2)
where: X = actual volatiles concentration, wppm
X
E
= equilibrium volatiles concentration, wppm
t = time, minutes
k = Diffusion coefficient, feet/minute
a = solid particle area, feet
2
/feet
3
Equation (2) can be transformed by conventional mathemat-
ics to one that can be numerically integrated as shown
below.
dX/(X-X
E
) = - k*a*dt
(3)
The next few paragraphs describe the components of this
relationship.
Refinements of components in
Equation 3
If the equilibrium between the solid and the volatile are
such that the equilibrium content of the volatile in the solid
approaches zero, equation (3) can be integrated directly to
give equation (4) shown below:
X
f
/X
o
= e
-ka
(4)
where: X
F
= the final concentration of the volatile in the
solid.
Xo = the original concentration of the volatile in the solid.
However, it is rare that the equilibrium concentration ap-
proaches zero. Thus the assumption that Xe = zero is a
dangerous assumption and will almost always lead to an
inadequate design. It is imperative that equilibrium be con-
sidered in designing a volatiles removal system. Techniques
for determining the equilibriumwill be discussed in the next
section of this study.
As indicated in equation (2), the rate of drying depends
on both the amount of surface area available and the mass
transfer coefficient as well as the difference between the
actual and equilibrium concentration. The standard aca-
demic assumption is that the solid being dried exists as a
single non-porous cylindrical or spherical particle. With this
assumption, the overall mass transfer coefficient depends
on effective particle radius, diffusion rate through the solid
and the film mass transfer coefficient between the solid
surface and the gas phase.
This idealised situation rarely exists for the following
reasons:
• Volatiles stripping applications often involve solids that
are highly porous with high surface areas.
• ‘Bed characteristics’ as well as the individual particles,
limit the overall mass transfer coefficient.
The nature of both the drying equipment and the morpholo-
gy of the solid being driedmakes it mandatory that the mass
transfer coefficient be determined either experimentally or,
if sufficient data exists, from an empirical correlation. This
can best be accomplished by lumping the two parameters
(k and a) together into an overall mass transfer coefficient
(K) and modifying equation (3) as follows:
dX/(X-Xe) = -K*dt
(5)
where: K= The lumped parameter with units of 1/minutes.
An inspection of equation (5) indicates that the equipment
residence time can be determined by numerical integration
of the equation if ‘K’ is known and an equilibrium relation-
ship between the solid and the volatile exists. Development
of an equilibrium relationship will be discussed in the next
installment and the outline of a spreadsheet to do the
calculations will be provided after that in the third part of
this study.
Theoretical relationships
R =- K*DF
(1)
where: R = the rate of volatiles removal
K = a constant which depends on solidmorphology
and type of drying equipment.
DF = the driving force for removal of volatiles from the solid.
This is usually the difference between the actual concentra-
tion and the equilibrium concentration.
dX/(X-X
E
) =- k*a*dt
(3)
where: X = actual volatiles concentration, wppm
X
E
= equilibrium volatiles concentration, wppm
t = time, minutes
k = Diffusion coefficient, feet/minute
a =solid particle area, feet
2
/feet
3
inlet wet solid
inert gas
off gas
Purge Bin
stripped solid
Figure 1: Purge bin relationships