Page 198 - Separation process principles 2

5.2 Solid-Liquid Cascades 163

developed in subsequent chapters. As will be seen, single Assuming equilibrium, the concentration of soluble ma-
cannot be obtained from rigorous models because terial in the overflow from each stage is equal to the concen-
of the nonlinear nature of rigorous models, malung computer tration of soluble material in the liquid part of the underflow
calculations a necessity. from the same stage. Thus,

5.2 SOLID-LIQUID CASCADES
In addition, it is convenient to define a washing factol; W, as
Consider the N-stage, countercurrent leaching-washing
process shown in Figure 5.3. This cascade is an extension of
the single-stage systems discussed in Section 4.7. The solid
feed, entering stage 1, consists of two components A and B,
If (5-I), (5-2), and (5-3) are each combined with (5-4) to
of mass flow rates FA and FB. Pure liquid solvent, C, which
eliminate Y, and the resulting equations are rearranged to
can dissolve B completely, but not A at all, enters stage N at
allow the substitution.of (5-3, the following equations result:
a mass flow rate S. Thus, A passes through the cascade as an
insoluble solid. It is convenient to express liquid-phase con-
centrations of B, the solute, in terms of mass ratios of solute
to solvent. The liquid oveg7ow from each stage, j, contains 5
mass of soluble material per mass of solute-free solvent, and
no insoluble material. The underJlow from each stage is a
slurry consisting of a mass flow FA of insoluble solids, a
constant ratio of mass of solvent/mass of insoluble solids
equal to R, and 4 mass of soluble materiallmass of solute-
free solvent. For a given solid feed, a relationship between
the exiting underflow concentration of the soluble compo- Equations (5-6) to (5-8) constitute a set of N linear alge-
nent, XN, the solvent feed rate, and the number of stages is braic equations in N unknowns, X,(n = 1 to N). The equa-
derived as follows. tions are of a tridiagonal, sparse-matrix form, which-for
If equilibrium is achieved at each stage, the overflow example, with N = 5-is given by
solute concentration, Yj, equals the underflow solute concen- -
tration in the liquid, Xi, which refers to liquid held by the 1 - 1 0 0 0
solid in the underflow. Assume that all soluble material, A, is () - ( ) 1 0 0
dissolved or leached in stage 1 and all other stages are then
washing stages for reducing the amount of soluble material 0 (d) -(+) 1 0
lost in the underflow leaving the last stage, N, and thereby
0 0 (f) - ( 1
increasing the amount of soluble material leaving in the
overflow from stage 1. By solvent material balances, it is - 0 0 0 (dl -(+)
readily shown that for constant R, the flow rate of solvent
leaving in the overflow from stages 2 to N is S. The flow rate
of solvent leaving in the underflow from stages 1 to N is
RFA. Therefore, the flow rate of solvent leaving in the over-
flow from stage 1 is S - RFA.
A material balance for the soluble material around any
interior stage n from n = 2 to N - 1 is given by

Equations of type (5-9) can be solved by Gaussian elirnina-
For terminal stages 1 and N, the material balances on the sol- tion by starting from the top and eliminating unknowns X1,
uble material are, respectively,
Xz, etc. in order to obtain

S
y1 *,Y"- Yn+r 1 YN-1 YN Solvent C
2 N-1 N *
insoluble A x~
Soluble B Figure 5.3 Countercurrent leaching
-
'A, F~ or washing system.