Bends differing radii

Example 10.7 An air duct of 2- by 2-ft square cross section turns a bend of radius 4 ft to the center line of the duct. If the measured pressure difference between the inside and outside walls of the bend is 1 in of water, estimate the rate of air flow in the duct. Assume standard sea-level conditions in the duct, and assume ideal flow around the bend. 

In some cases, it may become necessary to move the material in a different direction and not in a straight line. A smooth bend (long radius type) of the conveyor belt is recommended in such cases. 

Consider the design of a glass window for a vacuum chamber (Fig. 18.6). It is a circular glass disc of radius R and thickness f, freely supported in a rubber seal around its periphery and subjected to a uniform pressure difference Ap = 0.1 MPa (1 atmosphere). The pressure bends the disc. We shall simply quote the result of the stress analysis of such a disc it is that the peak tensile stress is on the low-pressure face of... 

The minimum bend radius for flexible hose varies according to the size and construction of the hose and the pressure under which the system operates. Current applicable technical publications contain tables and graphs showing the minimum bend radii for the different types of installations. Bends that are too sharp will reduce the bursting pressure of flexible hose considerably below its rated valve. 

Water is flowing in a horizontal pipe bend at a velocity of 10 ft/s. The radius of curvature of the inside of the bend is 4 in., and the pipe ID is 2 in. A mercury manometer is connected to taps located radically opposite each other on the inside and outside of the bend. Assuming that the water velocity is uniform over the pipe cross section, what would be the manometer reading in centimeters What would it be if the water velocity were 5 ft/s Convert the manometer reading to equivalent pressure difference in psi and Pa. 

This formally simple procedure is very difficult to perform, however. Because of radiation from the bend, the azimuthal propagation constant v to be found is complex. Since the bend radius of the waveguide is typically larger than the wavelength, the real part of v can be large, too. Moreover, a number of modes of each slice with very different values of their effective indexes are to be considered simultaneously. It causes very serious... 

To understand drainage we have to discuss the pressure inside the liquid films. At the contact line between liquid films, a channel is formed. This is called the Plateau border. Due to the small bending radius (rP in Fig. 12.18), there is a significant Laplace pressure difference between the inside of the compartment and the liquid phase. The pressure inside the liquid is significantly smaller than in the gas phase. As a result, liquid is sucked from the planar films into the Plateau s border. This is an important effect for the drainage of foams because the Plateau borders act as channels. Hydrodynamic flow in the planar films is a slow process [574], It is for this reason that viscosity has a drastic influence on the evolution of a foam. Once the liquid has reached a Plateau border the flow becomes much more efficient. The liquid then flows downwards driven by gravitation. 

E is the modulus of elasticity, d is the thickness of the plate, r is the radius of curvature, and v is the Poisson ratio (typically 0.3). For a discussion of Eq. (40) see Refs. 137 and 138. Spontaneous bending was used to calculate the difference in surface stress between opposite faces of several crystals, including InSb, GaAs, InAs. GaSb. and AlSb [134,139] and aluminum nitride [140]. 

We note that the elasticity discussed above is only for planar bilayers under compression or tension and does not extend to the bending of bilayers. On the contrary, our analysis has shown that the bending of a bilayer is favoured down to the critical packing radius (assuming that the lipids can freely rearrange by lateral movement and/or flip-flop), and that bending elasticity sets in only for radii smaller than this critical value. The elasticity of a fluid bilayer is therefore seen to be profoundly different from that of a classical elastic plate or shell. 

Schematic illustrations of the bending type specimen test set-up (main parts striker, fixture with anvils, specimen (instrumented for higher testing rates) are shown in Fig. 2. While an identical striker was used in all bending test set-ups, the anvil geometry and distance in the 2 test set-ups is different (anvil radius of 1 mm in the Charpy fixture anvil radius of 5 mm in tlie SENB fixture). For tensile type fracture specimens 2 test set-ups were realized, to accommodate either the pin-loaded C(T), SENT and DENT specimens (Fig. 3(a)) or the grip loaded CRB specimens (Fig. 3(b)). While conventional grips were used for testing rates up to 0.1 m/s, a modified C(T) or CRB fixture was used with a slack adapter at higher testing rates (for more details see reference [4]).

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