Equations of Distillation Modelling

The basic equations below fully describe a distillation column. These equations define the overall column total material balances, energy balances, and product compositions. Internal to the column, they describe equilibrium conditions, internal (stage-to-stage) component and total material balances, and internal energy balances. The independent variables of a column are the product rates and compositions, internal vapour and liquid rates and compositions, and stage temperatures. Equilibrium constants, also called K values, and mixture enthalpies are dependent vari ables. Each stage is assumed to be at equilibrium (a theoretical stage), though an efficiency can be applied in the equations.

The equations were first referred to as the MESH equations by Wang and Henke (1966). The MESH acronym stands for:

Material or flow rate balance equations, both component and total.

Equilibrium equations including the bubble and dew point equations.

Summation or Stoichiometric equations or composition constraints.

Heat or enthalpy or energy balance equations.

The MESH variables are referred to as state variables. These are:

• Stage temperatures, Tj

• Internal total vapour and liquid rates, Vj and Lj

• Stage compositions, yji and Xji, or instead, component vapour and liquid rates, vji and j

The equilibrium equation is: material withdrawn, wpi, is subtracted from the com ponent material balance. By convention, material yji = Kjlxjl or Vji/Vj = Kjllji/Lj leaving a tray has a negative value and material enter ing a tray has a positive value. The equilibrium constant or K-value, Kji, can be The total material balances are organized in the a complex function itself, dependent on the composi- same manner as the component balances. The total tions, Xji and yji material balance for the simple stage of Figure 2 is:

K i = K i(Tj, Pj, Xj i, yji) Vj +1 + Lj_1 - Vj - Lj = 0

The dependence of Kyi on Xji and yji often appears in The same convention applies to feed and product the MESH equations. The component rates can also trays where the total flow rate of a feed, Ff, is positive be expressed in the terms of each other, giving: and the product, Wp, is negative.

The equilibrium equation and the composition vji = lji(KjiVj/Lj) = jSji constraint are combined to get the bubble point equa tion:

and the dew point equation:

KjVj/Lj is termed the stripping factor, j while

Lj/KjiVj is termed the absorption factor, Aji. 1 C v)i

The summation equation or composition con- L ^ * = K _ 1 _ 0

straints simply states that the sum of the mole frac- ' =

tions on each stage is equal to unity. For the liquid These, or some variation, are important in some phase: methods to find the stage temperature, especially for more narrow boiling mixtures.

L x.. _1=0 or L l i/L —1=0 or The energy balance equations are required in any

, = 1 , = 1 rigorous method. In narrow-boiling mixtures, they c influence the internal total flow rates. In wide-boiling

L yji/Kji —1=0 mixtures and in columns where there are great

' = 1 heat effects (e.g. oil refinery fractionators) they also

, . , , strongly influence stage temperatures. The overall and for the vapour phase:

energy balance for a column with one feed and side

L yji — 1 = 0 or L Vji/Vj — 1=0 or '"1 '"1

product is:

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