Leaching is the preferential solution of one or more compounds from a solid mixture by contact with a liquid solvent. The solvent partially dissolves the solid material so that the desired solute can be carried away.
Typical users include:
Liquid-solid phase equilibrium is important in understanding leaching, crystallization, and adsorption. Diffusion through solids is slow, even through pores in the substance, and so equilibrium is harder to achieve.
Separation of a solid phase from a liquid phase is done by sedimentation, filtration, or centrifugation. Complete separation is essentially impossible, so must deal with some degree of liquid entrainment on any "wet" solid phase.
Leaching can be batch, semibatch, or continuous. It usually operates at an elevated temperature to increase the solubility of the solute in the solvent. Calculations involve three component (solid, solvent, solute) systems.
Feed to a leaching system typically is solid, consisting of basically insoluble carrier material and a (usually desirable) soluble compound. The feed usually must be prepared by grinding or chopping. It is then mixed with a liquid solvent. The desired material dissolves (to some extent) and so leaves when the liquid is drawn off as overflow. The solids are then removed as underflow. The underflow is wet, and so some of the solvent/solute mixture is carried out here as well.
Flow through a leaching system may be crosscurrent or countercurrent.
Modeling a leaching system requires several assumptions to make the system "ideal".
The solubility of solute may have an upper bound, limiting how much solute the solvent may hold. Ideally, the carrier will not dissolve, and so will not be present in the overflow. This is a generally safe assumption, although allowance should be made for entrainment of solid flakes, etc., in the overflow from the first stage.
Mixing of the solid and the solvent is critical. Typically, "perfect mixing" is assumed, as is the idea of an equilibrium stage (the solid and liquid phases on each stage are in equilibrium). These assumptions imply that all the liquid within the stage has the same composition and so the overflow and the liquid carried in the underflow are identical. This is known as the uniform solution assumption and will result in a linear equilibrium curve.
One also needs to determine how much liquid leaves entrained with the solids in the underflow. The simplest option is the assumption of constant solution underflow which means that every stage has the same, fixed ratio of solution to solid in the underflow stream. Like the equimolar overflow assumption in distillation, this will produce a linear operating curve. The first stage is again problematic, since it must "wet" the solid (fill pores, etc.) and so will typically pick up more liquid than subsequent stages.
More generally, the amount of solution in the underflow depends on the properties of the solution, which are dependent on its composition. The amount of solute present seems to effect the "stickiness" of the solution. As a result, "draining data" is typically collected. This relates the solution/solid ratio to the composition of the solution and is used to determine the nature of the operating curve.
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As has become customary in this course, we develop the operating equations from the balances describing the system. The steady state material balance over any stage in a countercurrent flow system is written as:
The operating equation is obtained by rearranging the component balance:
In some circumstances, it may be useful to use the total material balance to eliminate Vn+1 from the equation:
In the general operating case, the density and viscosity of the solution change with the solute concentration and the mass retained by the solid phase changes from tank to tank. Consequently, the ratio of entrained liquid to solution varies from stage to stage. In this case, the operating curve is not linear.
In the special case of constant solution underflow, the mass retained by the solid does not depend on the concentration and the operating line is linear after the first stage.
Two main types of calculations are usually performed:
Leaching calculations are almost always based on the principle that the solid will not dissolve into the solvent. Thus, the concentrations x of solute in the solvent entrained in the slurry and y of the solute in the solvent liquid phase can be expressed on a solid free basis without significantly changing the calculation. The equilibrium behavior of the system establishes key behaviors.
Begin with the assumptions that each stage has enough contact time that the system can reach equilibrium, that there is enough solvent present in each stage to allow equilibrium removal of solute from solvent, and that the solute will not absorb on the solvent. The uniform solution assumption then applies, and the equilibrium x-y curve will be a straight line, xe=ye.
Two subsets of the uniform solution case should be noted. In the first, the solute is infinitely soluble in the solvent. In this case, all values of x and y from 0.0 to 1.0 may be obtained. In the second case, the solubility is limited to some maximum value, xs. The x-y diagram will be a straight line, but will not go all the way from 0 to 1. Instead, at xs it will become vertical.
If the conditions (contact time, adequate solvent mass, no adsorption) that determine uniform solution do not hold, the x-y diagram may be curved (often, it will look like that used in distillation).
In the case of constant solution underflow, both the equilibrium and operating curves are linear and it is possible to solve for the number of stages directly. The general solution is developed using the Kremser Equation is:
If solution underflow is not constant, the operating equation for leaching is not linear. In this case, an approach similar to the McCabe-Thiele method for distillation columns can be used.
Another approach to solving these problems uses a graphical method similar to the Ponchon-Savarit method for distillation columns; however, a composition-composition diagram is used for construction rather than an enthalpy-composition diagram. The method assumes that all streams are a mix of solid and solution and that the ratio of solid to solution can be calculated.
The working plot is constructed using modified compositions. Define a to be the mass (or concentration) of solute, b the mass of solid (zero for an insoluble carrier), and s the mass of solvent, then calculate:
The upper curve will be a line if solution underflow is constant. The lower curve will collapse to the X axis (Y=0) when the carrier is completely insoluble. The curves are connected by tie-lines, based on the relationship between the phases. If the uniform solution condition holds so that xe=ye, these will be vertical; if not, they will be slanted and must be determined. Generally speaking, the equilibrium data must be obtained experimentally.
Once the composition diagram is constructed, a "J point" may be determined. This is useful in some, but not all problems. J is defined as the in terms of the total amount of solute entering (or leaving, they're equal at steady state):
The "P point" is used in the constructions. It is given by the equations:
Operating lines are constructed by connecting an L point with the P point. The intersection with the axis will be the V point for the next stage.
Begin construction at Va point. First trace upward to the equilibrium curve along a tie line (remember!, the tie lines are vertical for when the uniform solution conditions hold). This will be the L1 point. Next construct a line by connecting L1 to P. The V2 point will lie at the intersection of this line and the X-axis (the operating curve). Trace up a tie line to L2, construct a line to find V3, etc. Continue to alternate between the tie lines and constructed operating lines until the Vb point has been passed. The number of triangles formed will correspond to the number of leaching stages required.
Original: 26 March 1997
Revised: 2 April 1997, 27 Feb 2003
Copyright 1997, 2003 by R.M. Price -- All Rights Reserved