Supercritical flow

For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile h(x). The dimensionless Fronde number is Fr = VVg/j. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c = V they cannot propagate upstream in supercritical flows. The specific energy Ejp is nearly constant. 

Hintermair, U. and Zhao, G. and Santini, C.C. and Muldoon, M.J. and Cole-Hamilton, D.J. (2007). Supported ionic liquid phase catalysis with supercritical flow. Chem. Commun., 14, 1462-1464. 

Fig. D-5 shows an external compression air-intake designed for optimized use at Mach number 2.0. Fig. D-6 shows a set of computed airflows of an external compression air-intake designed for use at Mach number 2.0 (a) critical flow, (b) sub-critical flow, and (c) supercritical flow. The pressures at the bottom wall and the upper wall along the duct flow are also shown. Two oblique shock waves formed at two ramps are seen at the tip of the upper surface of the duct at the critical flow shown in Fig. D-6 (a). The reflected oblique shock wave forms a normal shock wave at the bottom wall of the throat of the internal duct. The pressure becomes 0.65 MPa, which is the designed pressure. In the case of the subcritical flow shown in Fig. D-6 (b), the shock-wave angle is increased and the pressure downstream of the duct becomes 0.54 MPa. However, some of the airflow behind the obhque shock wave is spilled over towards the external airflow. Thus, the total airflow rate becomes 68% of the designed airflow rate. In the case of the supercritical flow shown in Fig. D-6 (c), the shock-wave angle is decreased and the pressure downstream of the duct becomes 0.15 MPa, at which the flow velocity is stiU supersonic.

Figure D-6. Comparison of experimental and theoretical airflows under three types of operational conditions for the air-intake shown in Fig. D-5 (a) critical flow, (b) subcritical flow, and (c) supercritical flow.

Values of y for supercritical flow of a gas (r < through orifices are given by Benedict [/. Basic Eng., 93, 121-137 (1971)]- For the case of liquids, expansion factor y is unity, and Eq. (10-27) should be used, since it allows for any difference in elevation between the upstream and downstream taps. 

This result may be obtained independently by differentiating Eq. (10.111) with respect toy and equating to zero. It may be observed that the depth, which may be plotted vertically to determine the curve, is also represented by the horizontal distance from the vertical axis to the 45° line. It is also seen that the upper limb of such a curve corresponds to subcritical flow, while the lower limb refers to the alternate condition of supercritical flow. 

By far the most important of the local nonuniform flow phenomena is that which occurs when supercritical flow has its velocity reduced to subcritical. We have seen in these example scenarios that there is no ordinary means of changing from lower- to upper-stage flow with a smooth transition, because the theory calls for a vertical slope of the water surface. The result, then, is a marked discontinuity in the surface, characterized by a steep upward slope of the profile, broken throughout with violent turbulence, and known universally as the hydraulic jump. 

The problem of determining where a hydraulic jump will occur is a combined application. In the case of supercritical flow on a mild slope, for instance, the tail water depth y2 is determined by the uniform flow depth jo for that slope. The rate of flow and the application of Eq. (10.133) then fix yu and the length of the M3 curve required to reach this depth from the upstream control may be computed from Eq. (10.123). Similarly, in the case of subcritical flow on a steep slope, the initial depth is equal to y0, the tail water depth is given by Eq. (10.133), and the length of the Si curve to the jump from the downstream control is computed from Eq. (10.123). For application of the hydraulic jump to design problems, and for analysis of the jump in circular and other nonrectangular sections, the reader is referred to more extensive treatises on the subject [42],... 

Knapp, R. T., Design of Channel Curves for Supercritical Flow, cited in R. L. Daugherty and A. C. Ingersoll, Fluid Mechanics, McGraw-Hill, New York, 1954. 

Figure 1. Supercritical flow reactor. Key (I) Mettler balance (2) flask with 1 0 (filtered and deaerated) (3) HPLC pump (4) bypass (three-way) valve (5) feed cylinder (6) weather balloon with feed solution (7) probe thermocouple (type K) (8) ceramic annulus (9) Hastelloy C-276 tube (10) entrance cooling jacket (11) entrance heater (12) furnace coils (13) quartz gold-plated IR mirror (14) window (no coils) (15) guard heater (16) outlet cooling jacket (17) ten-port dualloop sampling value (18) product accumulator (19) air compressor (20) back-pressure regulator and (21) outflow measuring assembly.

Figure 2. Single-pass supercritical flow apparatus.

M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. 

The careful selection and design of control valves is important good flow control must be achieved, while keeping the pressure drop as low as possible. The valve must also be sized to avoid the flashing of hot liquids and the supercritical flow of gases and vapors. Control valve sizing is discussed by Chaflin (1974). 

In preparative liquid-carbon-dioxide-based supercritical flow chromatography (SFC), smaller particles in the 5-10 /tm range are used due to the decreased viscosity of the mobile phase. The pore size of the particles should be large enough to allow the molecules to readily diffuse into and out of the pores. In reversed-phase HPLC, longer alkyl chains provide better load- ability because of the higher volume of interaction be-... 

Figure 1 is a schematic of one of the two supercritical flow reactors used in this work. The system is first brought up to the operating pressure by an air compressor. An HPLC pump forces the reactant solution through the reactor, the ten-port valve and dual-loop sampling system, and into the product accumulator, where the flow of products displaces air through a back-pressure regulator. The reactant inflow is rapidly heated to reaction temperature by an electric entry heater/water jacket combination, and maintained at isothermal conditions by a Transtemp Infrared furnace and an exit electric heater/water jacket combination. 

Astrakharchik-Farrimond E, Shekunov BY, York P, Sawyer NBE, Morgan SP, Somekh MG, See CW. Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry. Exp Fluids 2002 33 307-314. 

Shekunov BY, Hanna M, York P. Crystallization process in turbulent supercritical flows. J Cryst Growth 1999 198/199, 1345-1351. 

Clamen, a. Gauvdj, W. Ft. 1969 Effects of turbulence on the drag coefficients of spheres in a supercritical flow regime. AIChE Journal 15, 184-189. 

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