Determination of the polynomial defining underwood's equations in short-cut distillation design

The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood equations, which can also be applied to complex columns, are used to calculate it (e.g., several feeds and side products and side-stream strippers and enrichers). In this work, it is shown that Underwood's equations of an N-component mixture can be represented as a polynomial of degree N and general relations for its coefficients are derived. In addition, the advantage of solving Underwood's equations in a polynomial form is shown. Any conventional numeric method may be used to find the roots of this polynomial because it is continuous and smooth (the discontinuities are eliminated).