Hemispherical shells

Figure 28-5 Steel, agate, and boron carbide mortars and pestles. The mortar Is the base and the pestle Is the grinding tool. In regard to boron carbide, the mortar is a hemispheric shell enclosed in a plastic or aluminum body. The pestle has a boron carbide button at the tip of a plastic handle. [Courtesy Thomas Scientific, Swedesboro, NJ, and Spex Industries, Edison, NJ]...

Electrical bulk properties of ionic solids can be rather inhomogeneous (Sec. 3.1). In the following it is shown that microelectrodes are a very useful tool to gain spatially resolved information on the conductivity of such inhomogeneous solids. Let us first consider the case of a spherical microelectrode (radius rme) atop a sample with homogeneous bulk conductivity Ubuik- The bulk resistance R between the microelectrode and a hemispherical counter-electrode of radius rce (Fig. 12a) can be calculated by integrating the infinitesimal resistances of hemispherical shells according to... 

On integrating again over 6 (from 2ir to 0) and (tt/2 to 0) we obtain the total perpendicular component of momentum carried by molecules of speed Cj in the hemispherical shell dr, distant r from dS, with velocity vectors pointing toward dS ... 

At the end of 2 min, the fresh grout flowing through the initial shell reaches its gel time, and thin incipient gel hemispherical shells begin to form at the new grout-groundwater interface. Figure 13.15d shows the condition at 2 min. 

The resistance contribution of a finite hemispheric shell of radius r on the metal sphere of radius a is ... 

The forces acting on a hemispherical section of a soap bubble are the film tension force, lirrof, pulling vertically downwards (Fig. 1.13), and the force due to the excess pressure, p, acting vertically upwards on the hemispherical section of bubble, irr p. The weight of the hemispherical shell of fluid can be neglected compared with either of these forces. As the hemispherical section is in equilibrium these two forces must balance, thus... 

The resistance of an electrolyte, or the tissue surrounding the electrode, is represented by a simple resistor / e. In saline, ions can move freely, resulting in low value of around 50 2 cm. For a 10-mm electrode implanted in biological tissue, the resistance is typically around 500 1500 S2. Certain tissues (e.g., muscle fascia) can also have a small capacitive component, making the impedance slightly frequency dependent. It should be noted that while all the other electrode capacitance and resistance parameters scale with an relationship, the electrolyte resistance is inversely proportional to the electrode radius r. The resistance relationship for a hemispherical electrode can be derived by integrating the electrolyte resistance of a hemispherical volume from the electrode surface to infinity. Each slice of the hemispherical shell has an area of 2-jt-r, and a thickness dr. 

The assembly was composed of two separate units. The upper unit, comprising two hemispherical layers of uranium (with inner and outer radii of 12.25 cm and 14.0 cm, respectively) and one hemispherical graphite layer (with inner and outer radii of 14.0 cm and 17.2 cm, respectively) was laid on a steel diaphragm 0.4 cm thick. The lower (movable) unit incorporated the remainder of the parts. All hemispherical shells of uranium had cylindrical pole holes 2.2 cm in diameter. During measurements, these holes were plugged with specialized stoppers of core material. 

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