Within the two-phase envelope, one can draw another envelope representing the limits of supercooling of the vapor and superheating of liquid that can be observed in the laboratory; along each isotherm these are the points for which (∂P/∂ρ)_T = 0. Obtain this envelope for the Peng-Robinson equation, and plot it on the same figure as generated in problem 7.12. This is the spinodal curve. The region between the saturation curve and the curve just obtained is called the metastable region of the fluid. Inside the spinodal curve, the fluid is unconditionally unstable. The saturation curve is called the binodal curve. Outside, the fluid is entirely stable. It is possible to enter the metastable region with hot water by heating at atmospheric pressure in a very clean flask. Sooner or later, the superheated liquid becomes unstable, however. Describe what would happen to your flask of hot water under these conditions and a simple precaution that you might take to avoid these consequences.

Data from Problem 7.12

Plot P_r versus ρr for the Peng-Robinson equation with T_r = [0.7,0.9,1.0], showing both vapor and liquid roots in the two-phase region. Assume ω = 0.040 as for N₂. Include the entire curve for each isotherm, as illustrated in Fig. 7.1. Also show the horizontal line that connects the vapor and liquid densities at the saturation pressure. Use lines without points for the theoretical curves. Estimate T_r^sat by log(P_r^sat) = 2.333(1 + ω) (1 - 1/T_r^sat).

Transcribed Image Text:

8 00 7 6 LO 5 P (MPa) A 3 2 1 0 1.3 T₁ = 1.1 0.01 1.0 0.9 1 0.02 . I I 0.8 0.03 6 5 01 4 P (MPa) W 2 1 0 0 1.3/ / T₂ = 1.1 1.0 0.9 0.8 + 0.002 0.004 0.006 0.008 p(mol/cm³) p(mol/cm³) Figure 7.1 Comparison of the Ppt behavior of methane (left) and pentane (right) demonstrating the qualitative similarity which led to corresponding states' treatment of fluids. The lines are calculated with the Peng-Robinson equation to be discussed later. The phase envelope is an approximation sketched through the points available in the plots. The smoothed experimental data are from Brown, GG, Sounders Jr., M., and Smith, R.L., 1932. Ind. Eng. Chem., 24:513. Although not shown, the Peng-Robinson equation is not particularly accurate for modeling liquid densities.