System of Algebraic Equations of Distillation

The distillation process is described by a system of algebraic equations, for the deduction of which let’s consider a closed- loop covering, for example, the column overhead beginning from tray j (Figure 1.b).

Figure 1. (a) A distillation column with condenser, reboiler, and reflux capacity; (b) control volume (dotted line) for obtaining material balance equations for the top section.

The equation of component material balance:

Vj−1yi, j−1 = Lj xi j + DxiD ……….  ( 1 )

The equation of heat balance:

Vj−1Hj−1 = Ljhj + DhD + Qcon ………. ( 2 )

The equations of phase equilibrium (for “theoretical” tray):

yi, j = Ki j xi j ………. ( 3 )

The summation equations:

∑ yi j = 1, ∑ xi j = 1 ………. ( 4 )

Here, Ki j = f (T, P, x1 . . . xn, y1 . . . yn) is a coefficient of phase equilibrium.

Hj = ϕ(T, P, y1 . . . yn) and hj = ψ(T, x1 . . . xn) are the enthalpies of vapor and liquid, respectively.

At first sight, the system of Eqs. ( 1 ) ÷ ( 4 ) appears to be rather simple, but it is necessary to bear in mind that the equation of phase equilibrium [Eq. ( 3 )]

together with the equations of summation [Eq. ( 4 )] are always nonlinear, even in the case of the so-called ideal mixtures, with αih = Ki /Kh = const (the component relative volatilities are not influenced by temperature and composition).

In real mixtures, functions Ki, j have rather complicated form (especially for azeotropic and heteroazeotropic mixtures). Sometimes the system [Eq. ( 1 ) ÷ ( 4 )] is simplified with the rejection of the heat balance equation [Eq. ( 2 )] and with the adoption of the flows Lj and Vj constancy within each column section (the term section refers to the part of a column between the flow inlet and outlet points).

The system [Eq. ( 1 ) ÷ ( 4 )] may have a large number of equations. First, the number of theoretical trays N may be enormous. Second, number of components n may also be very large. For example, petroleum contains thousands of components, which actually, for practical reasons, will be combined into tens of pseudocomponents (or fractions).

The system [Eq. ( 1 ) ÷ ( 4 )] may be solved only by iteration, and the solution is not always immediately obtained, so it requires a high degree of initial approximation. As a result of the system [Eq. ( 1 ) ÷ ( 4 )] solution at the preset number of theoretical trays in each section, we get not only the compositions of products xiD and xi B, but also the compositions on all trays xi j and yi j

Referring to,
Petlyuk, F., B. (2004). Distillation Theory and Its Application to Optimal Design of Separation Units. Cambridge University Press: USA.