Equilibrium relationships, column distillation

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

The height of a distillation column depends on the feed conditions, the product purity specifications and the extent of separation through the vapour-liquid equilibrium relationship, but also on the type of tray or packing used in the column as this affects the rate of separation. Column vendors will normally provide information on tray or packing efficiencies.9,10... 

The height of an absorption column depends on the feed conditions, the product purity specifications, the solvent used and the extent of separation through the absorption equilibrium relationship, but also on the rate of separation. If the rate of mass transfer of the gaseous component from the gas phase into the liquid phase is slow, then the column needs to be longer to ensure that the required amount is removed. The rate of mass transfer depends on the mass-transfer coefficient, normally denoted kG or k. The value of the mass-transfer coefficient depends on the components in the gas feed and on the solvent used and is often determined experimentally. The type of packing used in the column will also have an impact on the column height as for distillation. 

The above applies to both binary and multicomponent distillation. In multicomponent distillation, once the above are specified, other components will distribute according to the equilibrium relationship. Frequently, a product spec sets the maximum concentration of impurities that can be tolerated in the product. Product specs are less than" specifications. The one impurity which is dependent on the column separation and is most difficult to achieve sets the composition specification in the column. This is illustrated in Table 3.1 for a propylene-propane separation (Ca splitter). Since the light nonkeys (hydrogen, methane, ethylene, ethane, and oxygen) end up in the distillate, their concentration in the distillate is independent of the column. Of the others, the most difficult purity to achieve sets the composition specification, Similarly, the heavy nonkeys (MAPD, C4 and... 

A distillation column is needed to separate methanol and water. The overhead stream should be at least 97% methanol and the bottom stream should contain no more than 3% methanol. Apply the graphical method to estimate the minimum number of stages to achieve this separation. Assume the equilibrium relationship, y = I3x, is linear and the operating line is the diagonal. 

The initial solvent and feed concentrahons, the desired hnal concentrations, and equilibrium behavior determine the direction of mass transfer, the minimum solvent-to-feed ratio, and the minimum theoretical tray requirements. These theoretical trays (or stages) are analogous in many respects to theoretical plates of a distillation colnmn, and absorption (or stripping) columns discussed by Fair in another chapter. For any particnlar solvent-to-feed ratio, equilibrium relationships and the operating line determine theoretical stage reqnirements. 

The calculation procedures [the 0 method, Kb method, and constant composition method] developed in Chap. 2 for conventional distillation columns are applied to complex distillation columns in Sec. 3-1. For solving problems involving systems of columns interconnected by recycle streams, a variation of the theta method, called the capital 0 method of convergence is presented in Secs. 3-2 and 3-3. For the case where the terminal flow rates are specified, the capital 0 method is used to pick a set of corrected component-flow rates which satisfy the component-material balances enclosing each column and the specified values of the terminal rates simultaneously. For the case where other specifications are made in lieu of the terminal rates, sets of corrected terminal rates which satisfy the material and energy balances enclosing each column as well as the equilibrium relationships of the terminal streams are found by use of the capital 0 method of convergence as described in Chap. 7. 

In the initial search, approximate solutions are obtained to the equations describing a distillation column see App. 9-2. This procedure makes use of component-material balances and equilibrium relationships, and it reflects fairly accurately the effect of varying kx and k2 on the separations bjdu bh/dh. In the initial search, the minimum value of 0 is determined for the specified value of the reflux ratio. Also in the initial search, the total distillate rate D is taken to be dependent on N, and it is estimated as described in App. 9-2. Thus, the function 0 is searched over only two variables, kx and fc2, as follows ... 

For highly nonideal solutions, the Almost Band Algorithm is recommended. The following formulation of the Almost Band Algorithm follows closely the one recommended by Izarraraz et al.7 Suppose that the column has a partial condenser and that the distillate rate Vt is specified. The equations required to describe the column consist of the equilibrium relationships... 

The nonlinear algebraic equations that describe a steady-state distiUalion column consist of component balances, energy balances, and vapor-liquid phase equilibrium relationships. These equations are nonlinear, particularly those describing the phase equilibrium of azeotropic systems. Unlike a linear set of algebraic equations that have one unique solution, a nonlinear set can give multiple solutions therefore, the possibility of multiple steady states exists in azeotropic distillation. 

Eor certain applications in the heterogeneous azeotropic distillation column, vapor-liquid-liquid equilibrium (VLLE) will occur in the top part of the column. The basic equilibrium relationship to use is to equate the ffigacities of all components in all vapor and liquid phases as in the following equation ... 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

Multicomponent distillations are more complicated than binary systems due primarily to the actual or potential involvement or interaction of one or more components of the multicomponent system on other components of the mixture. These interactions may be in the form of vapor-liquid equilibriums such as azeotrope formation, or chemical reaction, etc., any of which may affect the activity relations, and hence deviations from ideal relationships. For example, some systems are known to have two azeotrope combinations in the distillation column. Sometimes these, one or all, can be broken or changed in the vapor pressure relationships by addition of a third chemical or hydrocarbon. 

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