Nickalls

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Functions.Nickalls

The function solves the general cubic equation of state

\begin{equation*} f(x) = a \, x^{3} + b \, x^{2} + c \, x + d = 0 \end{equation*}

after (RWD Nickalls, 1996).

The equation is transformed into a reduced form

\begin{equation*} a \, z^{3} - 3 \, a \, \delta^{2} \, z + y_{N} = 0 \end{equation*}

with roots \(\left(z_1,z_2,z_3\right)\) given by

\begin{align*} z_{1} = & x_{1} - x_{N} \\ z_{2} = & x_{2} - x_{N} \\ z_{3} = & x_{3} - x_{N} \end{align*}

where

\begin{align*} \delta^{2} = & \frac{\left(b^{2}-3 \, a \, c \right)}{9 \, a^{2}} \\ h = & 2 \, a \, \delta^{3} \\ x_{N} = -\frac{b}{3 \, a} \\ y_{N} = f(x_{N}) \end{align*}

The new parameters allow the roots \(\left(\frac{z}{\delta}\right)_{i}\) of the reduced cubic to be tabulated in terms of the parameter \(\frac{y_N}{h}\). Then the roots of the original equation are obtained as

\begin{align*} x_{1} = x_{N} + \delta \, \left(\frac{z}{\delta}\right)_{1} \\ x_{2} = x_{N} + \delta \, \left(\frac{z}{\delta}\right)_{2} \\ x_{3} = x_{N} + \delta \, \left(\frac{z}{\delta}\right)_{3} \\ \end{align*}