Relevant Moments Research Articles

An investigation into the elongation rolling of hollow billets

Results of experiments investigating the rolling of hollows of pure lead and of a lead— alloy using a laboratory mill, and of hollows of various grades and wall thicknesses of steel using a commercial rolling mill, are presented and discussed. Both mills employ oblique rolls which have clearly defined entry, diameter reduction, and exit zones, and are instrumented for the recording of all relevant forces and moments. Results show a strong dependence of the roll forces on the degree of elongation—the ration of the cross-sectional area of the billet before and after rolling—which affords a convenient means of regulating the process. It is shown that the conditions leading to minimum roll force are not recommended for some grades of steel, owing to the essentially tensile stress state arising in the material between the rolls and the resultant liability to cracking. Based on the findings of the investigation, two improvements were suggested for the process: (i) a modified form of plug was introduce, to alter the pattern of straining of the material being rolled and promote the development of an essentially compressive stress state and (ii) the rolls were aligned with their axes converging, to reduce the degree of twisting imparted to the billet, which latter has been shown to result in the development of tensile stresses and is the cause of numerous defects in the rolling of steel of reduced capacity for plastic strain. By this means it has been possible to roll at much greater degrees of elongation, and at the same time secure improved surface properties in the product and a reduction in the number of rejects.

Linear Breakeven Analysis Under Risk

This paper provides an analytical and multivariate characteristic function based approach to linear breakeven analysis under risk. Explicit probability distributions for the breakeven points are given when all the relevant parameters are correlated. If analytical expressions for these distributions cannot be found, the multivariate characteristic function approach is suggested for determination of the relevant probability moments. A numerical example and a discussion of the risk implications on breakeven analysis studies are included as well.

Turbulent convection in water over ice

Measurements have been made of the vertical velocity and simultaneous temperature fluctuation at fixed points in turbulent, statistically steady convection in water over an ice surface. Convection was confined to a horizontal layer of constant depth that lay under a stably stratified layer. Measurements of vertical velocities less than 1 mm/s were accomplished using a laser-Doppler velocimeter with a frequency-shifted reference beam. Vertical profiles of all relevant one-point moments, including joint temperature-velocity moments, up to third order were obtained. In the stable layer internal gravity waves were observed, and certain aspects of their behaviour were found to agree with existing theoretical analyses. Typical r.m.s. values of the temperature and velocity fluctuations were nearly equal to the appropriate ‘convection’ scales in the convection layer while in the conduction layer immediately above the ice surface they were more nearly equal to the ‘molecular’ scales. All of the observations were consistent with the existence of buoyant plume-like masses which dominated the structure of the flow. An analysis of the data using a conditional averaging technique to determine the mean properties of these masses showed that below the stable layer they accounted for over 80% of the mean-square fluctuations in temperature and velocity.

Kinetic Theory Approach to Electrostatic Probes

A spherical electrostatic (Langmuir) probe in a slightly ionized plasma is studied from a kinetic theory point of view. The two-sided distribution function method of Lees, which embodies the Mott-Smith approach, is used. The velocity space is divided into two regions along the straight cone tangent to the spherical probe, and different distribution functions are defined in the two regions. On satisfying the two relevant moments of the distribution function (continuity and number density flux) three simultaneous ordinary nonlinear differential equations, which are appropriate to all values of the Debye length, collision mean free path and probe potential, are obtained for determining the ion and electron number densities, and the potential. These equations reduce to the usual linear flux equations when the mean free path is much shorter than the probe radius and the Debye length. The equations are first linearized and solved for the case of small probe potential. Explicit solutions are given for the current-voltage characteristics and the distributions of the number densities and the potential. The general case of arbitrary probe potential is also studied. Results for the characteristics and for the potential drop in the sheath are presented for some representative cases. Many of the results which are obtained do not appear in the original simplified Langmuir model. The modifications required to take into account the curvilinear orbits of the charged particles are discussed.