RMP Lecture Notes

Leaching

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:

the metals industry for removing mineral from ores (acid solvents)
the sugar industry for removing sugar from beets (water is solvent)
the oilseeds industry for removing oil from soybeans, etc. (hexane or similar organic solvents)

The basic concepts of leaching also apply in the environment, where materials can be leached out by rainwater and carried into the groundwater supply. A simple, everyday example of a leaching process is making your morning coffee.

Liquid-Solid Equilibrium

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.

Principles

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 Assumptions

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.

Operating Equations

Some equations in this document are being displayed using MINSE, a browser independent approach to displaying equations on the web. If the equations are not properly formatted by your brower, you need to RENDER EQUATIONS (select this link) by invoking Ping's MINSE polymediator. This will run a special program that should cause them to be formatted for viewing by your browser.

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:

Following McCabe et al. (1993), V is used for the liquid solvent phase and L for the entrained liquid phase. The subscripts indicate the stage in which the stream originates, with a used for the fresh feed and b for the final product. Similarly, the component balance is:

The operating equation is obtained by rearranging the component balance:

which can be shown to run through the points (x_a, y_a) and (x_b, y_b).

In some circumstances, it may be useful to use the total material balance to eliminate V_n+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.

Calculations

Two main types of calculations are usually performed:

The extent of leaching is determined, usually by balance calculations. Efficiency depends on contact timen and the liquid-solid separation efficacy. The limit on extent is imposed by equilibrium constraints.
The number of stages required to reduce the solute content to a specified value is determined. As with distillation, the relationship between the equilibrium and operating lines will be used to determine the number of stages.

Solution techniques are algebraic or graphical. Graphical solutions may be set up using the actual compositions or on a "solid-free" basis.

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, x_e=y_e.

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, x_s. The x-y diagram will be a straight line, but will not go all the way from 0 to 1. Instead, at x_s 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).

Constant Solution Underflow

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:

For leaching, the equilibrium line is x_e=y_e, so the equation becomes:

To use this, you need to make sure that L_a=L_n (that the flow of liquid entrained with the solid is the same entering and leaving the first stage -- constant solution underflow is specified starting with the first stage) or calculate the first stage separately.

Variable Solution Underflow

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.

Use system component balances to determine x_a and x_b.
Construct operating curve. In general, the operating curve will not intersect the equilibrium curve as it did for distillation.
- Select an x_n somewhere in the middle. This gives an equivalent L_n.
- Use L_n and a material balance to obtain V_n+1.
- Find y_n+1 by a component balance
- Plot (x_n, y_n+1) as a point on the operating curve.
- Repeat as needed to obtain the full operating curve. Often a single point is adequate.
Operating stages may then be stepped off as triangles between the operating and equilibrium curves in the same fashion as was used for distillation. Normally, you start with the dilute product composition.

Solid-Free Calculations

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:

mass solute/mass solution, and

mass insoluble/mass solution and plot Y vs. X. Since there are two phases (overflow and underflow), there will be two curves.

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 x_e=y_e, 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):

Graphically, J is the intersection of the line connecting V_a with L_b and that connecting V_b and L_a. The J point is not used in the actual construction of stages; it can be used for lever arm calculations.

The "P point" is used in the constructions. It is given by the equations:

P represents the "net flow" of material through the system, so that

Effectively, this point pretends that there are mixers on either end of the leaching process, combining both the L and V flows into a single stream. This is thus a fictitious value, and one coordinate must be negative.

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 V_a 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 V_b point has been passed. The number of triangles formed will correspond to the number of leaching stages required.

References:

McCabe, W.L., J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, 5th Edition, McGraw-Hill, 1993, pp. 614-623.
McCabe, W.L., J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, 6th Edition, McGraw-Hill, 2001, pp. 742-47.
Seader, J.D. and E.J. Henley, Separation Process Principles, John Wiley, 1998, pp. 198-201.
Treybal, R.E., Mass-Transfer Operations, 3rd Edition, McGraw-Hill, 1987 (reissue). pp. 717-761.

R.M. Price
Original: 26 March 1997
Revised: 2 April 1997, 27 Feb 2003