Equilibrium Stage Operations Equilibrium stage operations are based on principles of phase equilibrium. Two phases are mixed together. Some of the components will partition between the phases as the system tries to reach equilibrium. When the phases are separated, one is enriched with the solute and the other depleted. This combination of mixing, approach to equilibrium, and separation is called an equilibrium stage. The basic calculation for an equilibrium stage is the flash calculation that you learned in your multicomponent thermodynamics class. You should review flash calculations in general, and pay special attention to bubble point and dew point calculations. A single equilibrium stage can be used to make a separation. All that is needed is a heater or cooler and a vessel with enough space that liquid and vapor can disengage. These are typically called something like "flash drums" or "knock-out pots". Both are applications of flash distillation. Cascades The separation achieved by a flash often fails to meet process requirements. In practice, several equilibrium stages are connected in series to form a cascade. Many separation processes (distillation, extraction, etc.) are based upon cascades of equilibrium stages. Arranging stages into cascades allows more separation or less energy input than is possible in a single stage. The effectiveness of a cascade depends on how close the stages are able to approach equilibrium. This, in turn, depends on how good the mixing and mass transfer are within the stage. We will often talk about ideal stages -- the theoretical ideal (or perfect) stage -- where the two phases are in equilibrium when they leave. In reality, the contact stages we can build are not ideal, but calculations based on ideal stages are a useful approximation. To relate ideal stages to actual ones, we can apply a stage efficiency. Key to the success of an equilibrium stage operation is the use of two separable phases. The heavier is usually assigned the symbol L, the lighter V. For example: Distillation -- the L phase is liquid, the V phase vapor Leaching -- the L phase is entrained with a solid, the V phase a liquid solvent Extraction -- both phases are liquid; that resulting in the desired product, or extract, is usually the V phase, regardless of density. In this course, we will be pretty loose in how we deal with compositions -- your text will extend "concentration" (amount of substance per unit volume) to include mass and mole fractions. Normally, x will be used to symbolize the composition of the L phase, y the composition of the V phase. Stages can be arranged in three main ways: cocurrent, countercurrent, and crosscurrent. In a cocurrent arrangement, both streams (L and V) flow in the same general direction; stage 1 contacts the L feed with the V feed. In countercurrent flow, the fresh streams enter at opposite ends; the first stage the L phase enters is the last stage for the V phase. A crosscurrent arrangement typically allows one stream to flow through the cascade in series, while the other flows through the stages in parallel. The streams in an equilibrium stage process are numbered according to the stage they leave, thus L 1 flows from stage 1 to stage 2; L 2 from stage 2 to stage 3. I like to use L 0 to denote the fresh stream entering stage 1; however, MSH will use numbers only for the interior stages and renames the entering L phase L a and the L phase stream leaving the process L b . The V streams are numbered so that V a is on the same side of the process as L a . This doesn't really effect hand calculations, but in my opinion, renaming streams makes computer assisted calculation trickier. Most solution techniques for cascade separation systems rely on finding the intersection between two sets of equations: one describing the equilibrium, the other describing the operating conditions. The equilibrium "equation" may be expressed as a data table or a plot as well as an analytic equation. Equilibrium sets the ideal limit on the extent of separation -- an equilibrium stage operation can never purify a stream past its equilibrium concentration. The operating equations for a process are set by its mass and energy balances. These describe the actual amounts and compositions of material and degree of contact. They thus set the limit on the rate or amount of material that may be separated. When a solution satisfies both sets of equations -- equilibrium and operating -- it fits the process. Depending on the complexity of the unit process, the equations may be solved analytically, numerically, or graphically. In this course, we'll spend a fair amount of time working with graphical solutions -- primarily because these tend to help most students reach a better understanding of how the processes work; but be aware that mathematical solutions may be your best choice for the problems you try to solve. References: 1. McCabe, W.L., J.C. SMith, and P. Harriott, Unit Operations of Chemical Engineering (5th Edition), McGraw-Hill, 1993, pp. 495-505. R.M. Price Original: 1/5/98 Modified: 1/7/98, 1/31/2003 Copyright 1998, 2003 by R.M. Price -- All Rights Reserved RMP Lecture Notes
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