Roots of the Underwood’s equations in short-cut distillation from a companion matrix eigenvalues

The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood’s equations, which can also be applied to complex columns (e.g., several feeds and side products and side-stream strippers and enrichers) are used to calculate it. When distributed components other than the light and the heavy key are considered in the separation, the knowledge of all the roots of the Underwood’s feed equation is essential. However, the discontinuous form of the Underwood´s feed equation; makes the search of all roots a hard task. Using the fact that the Underwood’s equation can be transform into a polynomial form, in this work a companion matrix of the polynomial is presented, permitting the solution of the equation as a Generalized Eigenvalue Problem, arriving to a reliable and efficient method for the calculation of all roots of the Underwood’s equations. One main feature of the proposed approach is that not initial guesses are required to find all the roots of the Underwood’s equation. Open, robust and reliable software such as EISPACK can be used to routinely calculate the minimum reflux ratio by Underwood’s equations.