Critical slope

To overcome the uncertainty of the actual onset of ionization, among several others, [80] the critical slope method has been developed. [25,81] It makes use of the fact that from theory realistic values of IE are expected at the position of the ionization efficiency curve where the slope of a semilog plot of the curve is... 

Fig. 2.19. Ionization efficiency curve of argon plotted on a linear scale (a) and as semilog plot (b). Extrapolation of the linear portion of a gives erroneous IBs, whereas the x-position of the tangent of an empirical critical slope to the sertrilog plot yields accuracies of 0.05 eV. Reproduced from Ref. [25] by permission. American Chemical Society, 1948.

Barfield, A.F. Wahrhaftig, A.L. Determination of Appearance Potentials by the Critical Slope Method. J. Chem. Phys. 1964, 47,2947-2948. 

Figure 1 Free energies for unreconstructed surface f, and reconstructed surface f, vs slope s. The critical slope, sc, and the slope of the surface at step bunches, St, are given by Sc = ,/ , and St = The thick curve in (a) represents the free energy of a hypothetical system... 

The limiting critical slope y on the heterogeneous side must therefore be identified with the critical eigenangle coc [(11.130)] of the homogeneous side, namely,... 

On decreasing critical temperature in B is denoted by Tc. The critical slope 2 is the lowest value that still leads to a bounded steady-state temperature. By further decreasing of

i),... 

The C scenarios. These scenarios, with the anomalous condition atye = y0, have already been discussed. Needless to say, the critical slope profiles are not of frequent occurrence. 

According to Table 14-13 the sample size requirement is N=40 for a standardized deviation 4. By squared interpolation, we obtain N=42 (=40x (4/3.9) ). Thus in this example the sample size requirement with regard to testing for intercept deviation from zero is less demanding than that of testing for a critical slope deviation. 

From Table 14-14 we obtain N= 48 (= 69 X (1/1.2) ) (i.e., in this case about the same number as that obtained for testing the critical slope deviation). 

Figure 10.3.1 also suggests how x itself loses stability. As r increases beyond 1, the slope at x gets increasingly steep. Example 10.3.1 shows that the critical slope f x ) = -1 is attained when r = 3. The resulting bifurcation is called a flip bifurcation. 

The global tilt of the sample surface can easily be higher than the critical slope angle of 3° defined above, thus rendering impossible any stiffness analysis with strict application of the topography criterion. However, parallelization... 

Kao and Hwang (1979) criticized this approach as being extremely stringent because it adds to construction costs and capital expenses. They implied that it would be better to properly understand the critical slope rather than to use a rule of thumb. 

Kao and Hwang (1979) observed that the critical slope for glass beads and for sand occurred at 23°(42% slope) from the horizontal. For other substances such as coal and walnut shells, the Initial motion appeared to occur at the interface between the particle bed and the pipe wall. This suggested that the internal friction between irregularly shaped coarse particles was higher than the friction at the wall of the pipe. 

The critical slope Kao and Hwang (1979) defined was the value for initial particle motion. For sand or glass beads it was 27° 2°, and for coal and walnut shells it was 37° 2°. 

New models for inclined flows are appearing, such as the work of Matsouk (1996) on inclined two-layer models The correct design of inclined flow must be based on empirical data on the critical slope. The principles reviewed in this chapter apply to dredging and transporting sand, gravei, coai, steei shot, and rocks from SAG, rod, or ball mills, cyclone underflows, and tailings, etc., which often have particle sizes larger than 70 p,m (mesh 200). 

Kao, D. T. Y., and L. Y. Hwang. 1979. Critical slope for slurry pipelines. Paper presented at the Hydrotransport 6 Conference of the British Hydromechanical Research Association, Cranfield, England. 

Substituting Equation 6-47 into the DuBoys equation (6-28), the critical slope to initiate motion of a bed of small sand particles in an open channel is... 

A slurry mixture of fine particles and water has a specific gravity of solids 4.1 and is flowing in a partially filled rectangular launder 600 mm wide. Determine the critical slope if the launder is to flow one-third full. The density of the mixture is 1250 kg/m (2.42 slugs/ft ) and the dynamic viscosity is 20 mPa s (4.17 x 10 Ibf-sec/ft ). 

For water flow, the critical slope is expressed in terms of the critical depth and the Manning roughness factor as... 

It is very important to avoid blocking slurry lines. The profile of the pipeline must not include any steep gradients in excess of the critical slope. Slurry pipelines should be... 

Erosion control blankets are used on slopes and disturbed soils where mulch must be anchored and other methods such as crimping or tackifying are neither feasible nor adequate. They are used on steep slopes, generally steeper than 3 1, and slopes where erosion hazard is high their use is especially appropriate for critical slopes adjacent to sensitive areas, such as streams and wetlands, and disturbed soil areas, where planting is likely to be slow in providing adequate protective cover. 

V. Frette. Sandpile models with dynamically varying critical slopes, Phys. Rev. Lett. 70, 2762-2765 (1993). 

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