Chemical Technology May 2016

MINERALS PROCESSING AND METALLURGY

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: 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-Xe) = -K*dt (5) where: K= The lumped parameter with units of 1/minutes.

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)

= the final concentration of the volatile in the

where: X F

solid. Xo = the original concentration of the volatile in the solid.

Figure 1: Purge bin relationships

inlet wet solid

off gas

dX/(X-X E

) =- k*a*dt

(3)

where: X = actual volatiles concentration, wppm X E

Purge Bin

= equilibrium volatiles concentration, wppm t = time, minutes k = Diffusion coefficient, feet/minute a =solid particle area, feet 2 /feet 3

inert gas

stripped solid

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Chemical Technology • May 2016