Fluid linear programming

Julian BR (1994) Volcanic tremor nonlinear excitation by fluid flow. J Geophys Res 99 11859-11877 Julian BR, Foulger GR (1996) Earthquake mechanisms from linear-programming inversion of seismic-wave amplitude ratios. Bull Seismol Soc Am 86 972-980... 

In the open system, the CO2 fugacity (Fig. 23.3) varies linearly over the reaction path from about 75 to a value of one, as we prescribed, tracing a path of lower fugacity than predicted for the closed system. To maintain the lower fugacity, the program allows CO2 to escape from the fluid into the external gas buffer (see Chapter 14). The pH (Fig. 23.1) follows a path of higher values than in a closed system, reflecting the loss of CO2, an acid gas. 

Although the combined asymptotic density/linear temperature program is usually the best approach, such a gradient is difficult to achieve without computer control of the pump and oven. Even an asymptotic (or linear) density gradient alone is beyond the capability of some SFC pumps. When faced with such limitations, it is reasonable to use a linear pressure program instead. The curvature of the density pressure-isotherms for some supercritical fluids (e.g., pure CO2) is such that a linear increase in... 

Supercritical Fluid (Dense Gas) Chromatography/Extraction with Linear Density Programming, L. M. Bowman, Jr., M. N. Myers, and J. C. Giddings, Sep. Sci. Technol., 17, 271 (1982). 

The overall density of the mobile phase is one of the most important parameters used to optimize separations in SFC with density programming as common in SFC as temperature programming in GC and eluent composition in HPLC [5]. Capacity ratios, k decrease roughly linearly at higher densities with different slopes for different classes of compounds, thereby affording changes in selectivity [5]. A similar effect is seen for the supercritical fluid elution of analytes from octadecylsilica sorbents, as seen in Fig. 2 [6]. 

Quiet, easy hum of a motor. Tap tap tap of feet on a hard floor—an audio discrimination program went primary— two sets of feet, plus the level breathing of two people. Constant pulse of light pressure on the enhanced retinas below closed eyelids indicated linear movement, backed up by inner ear fluid motion estimated at a fast walking pace. Posture was level he was still lying on his bed. 

Kuepper, S. Grosse-Ophoff, M. Klesper, E. Influence of linear velocity and multigradient programming in supercritical fluid chromatography. J. Chromatogr. 1993, 629, 10. 

Pressure changes in SFC have a pronounced effect on the retention factor k and thus the retention time l. The density of a supercritical fluid increases rapidly and nonlinearly with increases in pressure. Such density increases cause a rise in the solvent power of the mobile phase, which in turn shortens elution time. For example, the elution time for hexadecane is reported to decrease from 25 to 5 min as the pressure of carbon dioxide is raised from 70 to 90 atm. An effect similar to that of temperature programming in GC and gradient elution in FfPLC can be achieved by linearly increasing the column pressure or by regulating the pressure to create linear density increases. Figure 29-2 illustrates the improvement in chromatograms realized by pres-... 

There are general programs that can solve these equations for simple systems, e.g. computational fluid dynamics, CFD. However, the equations can easily become very difficult if they include stiff non-linear equations and complex geometry that call for a structured approach. 

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