The graphical McCabe—Thiele methods studied in the previous sections for the design of distillation columns are also widely used to analyze the operation of an existing column. In this case the total number of trays in the column NT is fixed. The feed tray may also be fixed, or, if there are multiple feed points available on the column, it may be varied. These fixed-column problems are called "rating problems," as opposed to the "design problems," in which NT is calculated.
There are a variety of possible rating problems. The two most commonly encountered are (with NT and NF fixed):
—To determine the reflux ratio required to achieve specified product purities xB and xD
—To determine the product compositions that result from specified values of reflux ratio and distillate flow rate
Both of these calculations involve iterative, trial-and-error solution techniques. Basically one guesses a solution and sees if the stepping procedure produces exactly the same number of trays in each section as has been specified (see Figure 2.26).
Notice that in both of these problems, two variables must be specified to define the system completely. This magic number of two occurs again and again in distillation (see Section 4). It is often called the "degrees of freedom" of the system. Mathematically the two degrees of freedom are the result of subtracting all the constraining equations describing the system (mass, component, and energy balances; VLE equilibrium relationships; and specified variables) from the total number of system variables.
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