Above the critical point or far from the saturation curve, only one real root to the cubic equation exists. If we are using Newton’s method, we can check how many phases exist by trying the two different initial guesses and seeing if they both converge to the same root. If they do, then we can assume that only one real root exists. Find the compressibility factors for methane at the following conditions, and identify whether they are vapor, liquid, or supercritical fluid roots. Complete the table. Compare your results to Z-charts.

When Newton’s method is applied with an initial guess of zero, erratic results are obtained at these conditions. Explain what is happening, and why, by plotting F (Z) versus Z for each iteration.

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T₁. 0.9 0.9 1.1 Pr 0.77 0.05 2.00