Cylindrical column, stresses

The onset of powder motion in a hopper is due to stress failure in powders. Hence, the study of a hopper flow is closely related to the understanding of stress distribution in a hopper. The cross-sectional averaged stress distribution of solids in a cylindrical column was first studied by Janssen (1895). Walker (1966) and Walters (1973) extended Janssen s analysis to conical hoppers. The local distributions of static stresses of powders can only be obtained by solving the equations of equilibrium. From stress analyses and suitable failure criteria, the rupture locations in granular materials can be predicted. As a result, the flowability of granular materials in a hopper depends on the internal stress distributions determined by the geometry of the hopper and the material properties of the solids. 

The instruments available for direct measurements are mainly associated with two classes of devices. One class measures the torque or stress on an immersed vaned paddle in a bed of powder compressed at a known bulk density, whilst the other class is the mould or column failure method. The more practical mould devices are those based on the formation of a powder column in a cylindrical container (sometimes split into two halves) (see Section 1.7.3.2). The container walls are then removed to leave a freestanding cylindrical column of powder, which is then subjected to loads placed on the top of the column imtil the column fails. 

By virtue of its chemical and thermal resistances, borosilicate glass has superior resistance to thermal stresses and shocks, and is used in the manufacture of a variety of items for process plants. Examples are pipe up to 60 cm in diameter and 300 cm long with wall tliicknesses of 2-10 mm, pipe fittings, valves, distillation column sections, spherical and cylindrical vessels up 400-liter capacity, centrifugal pumps with capacities up to 20,000 liters/hr, tubular heat exchangers with heat transfer areas up to 8 m, maximum working pressure up to 275 kN/m, and heat transfer coefficients of 270 kcal/hz/m C [48,49]. 

In order to measure bulk strength in the absence of confinement, as relevant to the stress conditions on the underside of an arch, a failure test is carried out after the formation stress is removed. This test reflects the failure conditions on the surface of an arch subjected to passive wall pressures generated by a mass flow hopper, and is measured by an unconfined failure test, as shown in Fig. 1.4. A column of material is compacted in a cylindrical cell and then subjected to axial loading after removal of the cell walls. This is a delicate operation, due to the sensitive nature of the samples and the effect of wall friction opposing the compacting forces. Frictional effects rapidly magnify with the length of... 

Joint panel shear failure joint shear capacity Vjh (corresponding to the average principle tensile stress pt = 0.42Vfcm (Priestley 1997), with fcm average concrete cylindrical compressive strength) is reached for column shear Vc = 42.8 kN ... 

The spherical type bubble column bioreactor (Figure 2.8) is another modified airlift reactor with spherical shape instead of cylindrical. Ihrough the addition of small amounts of agar to the culture medium, low shear stress is achieved as well as high oxygen transfer rates. As a result, 5.6 g.L of BC were produced after 72 hours of cultivation [54,55]. 

In a recent study Jakobsen et al. [71] examined the capabilities and limitations of a dynamic 2D axi-symmetric two-fluid model for simulating cylindrical bubble column reactor flows. In their in-house code all the relevant force terms consisting of the steady drag, bulk lift, added mass, turbulence dispersion and wall lift were considered. Sensitivity studies disregarding one of the secondary forces like lift, added mass and turbulent dispersion at the time in otherwise equivalent simulations were performed. Additional simulations were run with three different turbulence closures for the liquid phase, and no shear stress terms for the gas phase. A standard k — e model [95] was used to examine the effect of shear induced turbulence, case (a). In an alternative case (b), both shear- and bubble induced turbulence were accounted for by linearly superposing the turbulent viscosities obtained from the A — e model and the model of Sato and Sekoguchi [138]. A third approach, case (c), is similar to case (b) in that both shear and bubble induce turbulence contributions are considered. However, in this model formulation, case (c), the bubble induced turbulence contribution was included through an extra source term in the turbulence model equations [64, 67, 71]. The relevant theory is summarized in Sect. 8.4.4. 

A review article gives a thorough review of creep instability of steel and other metal shells and pressure vessels at elevated temperatures. Some of the same analyses can be adopted here for polymers at lower temperatures, for example, room temperature. For example, consider the axisymmetric buckling mode of a cylindrical shell under compression. (The beam column mode can be included under column buckling. Section 4.2.) For the axisymmetric mode, where axisymmetric buckling waves occur for RJh < 33, the critical buckling stress in compression is given by... 

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