The chaotic behavior of fixed-point methods for steady-state process simulation is studied. It is shown that direct substitution and Newton's method exhibit all of the rich structure of chaos (period doubling, aperiodicity, fractal basin boundaries, and related properties) on simple process examples. These examples include finding roots to the SoaveRedlich-Kwong and Underwood equations, dew point and flash calculations for heterogeneous mixtures, and a simple process flowsheet.For single variable problems, it is shown that direct substitution follows a classical period-doubling route to chaos. On the other hand, the chaotic behavior of direct substitution and Newton's method on multivariable problems is considerably more complex, and can give the appearance that no organized route to chaos is followed. For example, for the dew point problems, truncated period doubling, odd periodic cycles, and chaotic behavior can be observed, within which are embedded narrow regions of global convergence. Many numerical results and geometric illustrations are presented.

Experimental verification of theoretical investigations into the behaviour of liquid metal convective flows is required to validate analytical models and numerical simulation codes. A real-time radioscopic density visualization system has been developed and is beginning to provide useful data. The X-ray facility for studying opaque low-Prandtl-number fluid flow is described. Density changes as low as 10−3 g cm−3 can be visualized by careful control of scatter radiation. The new capability is demonstrated with natural convection benchmark experiments in a narrow vertical layer of gallium melt of aspect ratio A=1.4. The density field in that cell is three-dimensional. Good agreement between calculations and experiments is obtained. Experiments with solidification are provided for further theoretical use.

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