Stationary flow definition

In the coordinate-space treatment of TST, certain assumptions must be made concerning the nature of the Hamiltonian of the system. First, it must be assumed that it can be partitioned into the sum of two terms, the kinetic and the potential energy. Furthermore, one must also assume that the kinetic energy is positive definite and is quadratic in the momenta. With these assumptions, then the point of stationary flow in phase space and the saddle point of the potential energy... 

As we noted above, the kinetic energy is positive definite. Furthermore, it is quadratic in the momenta. As a consequence, we can reduce the search for points of stationary flow in phase space to one of finding the stationary points of the potential energy surface. To see how this comes about, consider the Hamilton s equations for the three velocities... 

Per definition, only the tensile strength oz and no inclination depends directly on the consolidation pre-histoiy obtained from a Taylor series linearisation of the yield locus [28] near Mohr circle of cohesive stationary flow, see Eq.( 9). Now the simplest formulation of the linear yield locus dependent on radius (Jvr ( R,st and center stresses (average pressure in the powder) ctvm [Pg.78]

The angle of internal friction, a, is defined as the equilibrium angle between flowing particles and bulk or stationary solids in a bin. Figure 4 illustrates the definition. The angle of internal friction is greater than the angle of repose. 

Though it is easy to determine formally what is a yield stress, in practice its measurement faces essential methodical and principal difficulties. Here two approaches are basically used, stationary and dynamic . During stationary measurements a flow curve is measured and interpretation of the results obtained leads to the definition of a yield stress. An example of such an approach is given in Fig. 1, where experimental points are shown conventionally. They can be obtained under the con-... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

As mentioned in Section 11.8.4, the parameters that are most important for a qualitative analysis using most GC detectors are retention time, tR adjusted retention time, t R and selectivity, a. Their definitions were graphically presented in Figures 11.16 and 11.17. Under a given set of conditions (the nature of the stationary phase, the column temperature, the carrier flow rate, the column length and diameter, and the instrument dead volume), the retention time is a particular value for each component. It changes... 

The science that deals with the deformation and flow of matter is called rheology. An important rheological concept is the shear force, sometimes called the shear stress, or the force that causes a layer of a fluid material to flow over a layer of stationary material. The rate at which a layer of a fluid material flows over a layer of stationary material is called the shear rate. A fluid flowing through a tube, for example, would be the fluid material, while the tube wall would be the stationary material. An important rheological measurement that is closely related to the resistance to flow is called viscosity. The technical definition of viscosity is the ratio of shear stress to shear rate ... 

Liquid chromatography (LC) and, in particular, high performance liquid chromatography (HPLC), is at present the most popular and widely used separation procedure based on a quasi-equilibrium -type of molecular distribution between two phases. Officially, LC is defined as a physical method... in which the components to be separated are distributed between two phases, one of which is stationary (stationary phase) while the other (the mobile phase) moves in a definite direction [ 1 ]. In other words, all chromatographic methods have one thing in common and that is the dynamic separation of a substance mixture in a flow system. Since the interphase molecular distribution of the respective substances is the main condition of the separation layer functionality in this method, chromatography can be considered as an excellent model of other methods based on similar distributions and carried out at dynamic conditions. 

If, in an infinite plane flame, the flame is regarded as stationary and a particular flow tube of gas is considered, the area of the flame enclosed by the tube does not depend on how the term flame surface or wave surface in which the area is measured is defined. The areas of all parallel surfaces are the same, whatever property (particularly temperature) is chosen to define the surface and these areas are all equal to each other and to that of the inner surface of the luminous part of the flame. The definition is more difficult in any other geometric system. Consider, for example, an experiment in which gas is supplied at the center of a sphere and flows radially outward in a laminar manner to a stationary spherical flame. The inward movement of the flame is balanced by the outward flow of gas. The experiment takes place in an infinite volume at constant pressure. The area of the surface of the wave will depend on where the surface is located. The area of the sphere for which T = 500°C will be less than that of one for which T = 1500°C. So if the burning velocity is defined as the volume of unbumed gas consumed per second divided by the surface area of the flame, the result obtained will depend on the particular surface selected. The only quantity that does remain constant in this system is the product u,fj,An where ur is the velocity of flow at the radius r, where the surface area is An and the gas density is (>,. This product equals mr, the mass flowing through the layer at r per unit time, and must be constant for all values of r. Thus, u, varies with r the distance from the center in the manner shown in Fig. 4.14. 

The term A is related to the flow profile of the mobile phase as it traverses the stationary phase. The size of the stationary phase particles, their dimensional distribution, and the uniformity of the packing are responsible for a preferential path and add mainly to the improper exchange of solute between the two phases. This phenomenon is the result of Eddy diffusion or turbulent diffusion, considered to be non-important in liquid chromatography or absent by definition in capillary columns, and WCOT (wall coated open tubular) in gas phase chromatography (Golay s equation without term A, cf. 2.5). 

The models described above are definitely stationary, while the electron cloud formation is not. It is assumed that the electron cloud does not evolve during the acceleration process and is not affected by the ions flowing through it. The latter condition requires that the number of the ions that are accelerated is much smaller than the number of hot electrons, Ni -C Ne. 

Rohrschneider [205,210] has developed a scheme for the characterization of stationary phases for gas chromatography. The scheme is based on the retention index (/). The retention index is a dimensionless retention parameter, designed to be independent of flow rate, column dimensions and phase ratio. The retention index of a solute is defined as 100 times the number of carbon atoms in a hypothetical n-alkane, which shows the same net retention time as that solute. This definition is illustrated in figure 2.2. By plotting the logarithm of the net retention time against the number of carbon atoms in n-alkanes, a straight line is obtained. The net retention time for a solute may then be located on the vertical axis, and the retention index found on a horizontal scale, which represents 100 times the scale for na... 

Since a definite function 82S leads to the stability condition, it operates as a Lyapunov function, and assures the stability of a stationary state. As the entropy production is the sum of the products of flows J and forces X, we have... 

Elucidation of the non-local holistic nature of quantum theory, first discerned by Einstein [3] and interpreted as a defect of the theory, is probably the most important feature of Bohm s interpretation. Two other major innovations that flow from the Bohm interpretation are a definition of particle trajectories directed by a pilot wave and the physical picture of a stationary state. 

A gravity dryer consists of a stationary vertical, usually cylindrical housing with openings for the introduction of solids (at the top) and removal of solids (at the bottom). Gas flow is through the solids bed and may be cocurrent or countercurrent and, in some instances, cross-flow. By definition, the rate of gas flow upward must be less than that required for fluidization. 

In the same way, the separation of cytochrome-c and lysozyme was achieved in 1988 by Ito and Oka [4] using the type J (or HSCCC) device. Tlie chosen ATPS system consisted of 12.5% (w/w) PEG-1000 and 12.5% (w/w) dibasic potassium phosphate in water. The two peaks were resolved in 5 h using a 1-mL/min flow rate, but the retention of the stationary phase was as low as 26%. The limitation of this type of apparatus is definitely the low retention of the stationary phase for ATPS systems. 

If the mobile phase is incompressible, as in liquid chromatography (LC), the dead volume (as so far defined) will be the simple product of the exit flow rate and the dead time. However, in LC, where the stationary phase is a porous matrix, the dead volume can be a very ambiguous column property and requires closer inspection and a tighter definition. 

In flow systems, one is frequently interested in the velocity of a given species with respect to v rather than with respect to stationary coordinates. This leads to the definition of the diffusion velocities ... 

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