Stress in Flat Plates

The design equations used to determine the thickness of flat ends are based on the analysis of stresses in flat plates Section 13.3.5. 

For a single circular opening in a flat plate with infinite boundaries in two directions (not ffirough the thickness) that is subjected to applied forces and stresses along opposite edges of the plate, stresses are increased above the nominal applied stress in the unperforated plate. The stresses decrease away from the opening until the nominal stress in the plate is obtained. The ratio of the stress at the examined point divided by the nominal stress is the stress intensity factor. 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

A layer of water is flowing down a flat plate that is inclined at an angle of 20° to the vertical. If the depth of the layer is 1 /4 in., what is the shear stress exerted by the plate on the water (Remember Stress is a dyad.)... 

A number of techniques have been developed to measure melt viscosity. Some of these are listed in Table 3.8. Rotational viscometers are of varied structures. The Couette cup-and-bob viscometer consists of a stationary inner cylinder, bob, and an outer cylinder, cup, which is rotated. Shear stress is measured in terms of the required torque needed to achieve a fixed rotation rate for a specific radius differential between the radius of the bob and the cup. The Brookfield viscometer is a bob-and-cup viscometer. The Mooney viscometer, often used in the rubber industry, measures the torque needed to revolve a rotor at a specified rate. In the cone-and-plate assemblies the melt is sheared between a flat plate and a broad cone whose apex contacts the plate containing the melt. 

When heat and mass are transferred simultaneously, the two processes interact through the Gr and Gq terms in Eq. (10-12) and the energy and diffusion equations. Although solutions to the governing equations are not available for spheres, results should be qualitatively similar to those for flat plates (T4), where for aiding flows (Gr /Gq > 0) the transfer rate and surface shear stress are increased, and for opposing flows (Gr Gq < 0) the surface shear stress is predicted to drop to zero yielding an unstable flow. 

The drive system of a rotational rheometer is likely to be optimized in one of two ways depending upon its preferred mode of operation. The most common form of rheometer is a controlled-rate (controlled-speed) device. This configuration is also used in most viscometers and has been around for decades. A shear rate is applied to a rotor by the motor controlling the viscometer s speed. The rotor is normally a flat plate or cylindrical cup. The stator is thus a cone or plate for the first two geometries or a cylindrical bob for the third (Figure HI. 1.1). The stator is linked to the rotor via the sample, which acts to couple the input signal like an automobile transmission. Thus, the torque on the stator when measured by a transducer is used to derive the shear stress in the sample. 

When parts are to be used in tension, the polyurethane should be bonded onto a tag rather than a flat plate. This helps eliminate stresses on the bonded surface. 

In the cone and plate rheometer, a cone-shaped bob is placed against a flat plate so that the fluid to be studied may be placed into the gap between the lower face of the cone and the upper face of the plate. Again, in the Searle method, the cone is rotated while in the Couette method the plate turns. In each case, the torque on the cone is measured. Figure 6.5 shows a Searle-type cone and plate arrangement. For this arrangement the shear stress is given by ... 

Air flows at a velocity of 25 m/s ova a wide flat plate that is aligned with the flow. The mean air temperature in the boundary layer is 30°C. Plot the mean velocity distribution near the wall in the boundary layer assuming that flow is turbulent and that the wall shear stress is given by ... 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

The analogy solutions discussed in the previous section use the value of the wall shear stress to predict the wall heat trans er rate. In the case of flow over a flat plate, this wall shear stress is given by a relatively simple expression. However, ir, general, the wall shear stress will depend on the pressure gradient and its variation has to >e computed for each individual case. One approximate way of determining the shear stress distribution is based on the use of the momentum integral equation that was discussed in Chapter 2 [1],[2],[3],[5]. As shown in Chapter 2 (see Eq. 2.172), this equation has the form ... 

A most ingenious use for soap films has been discovered by Griffith and Taylor.1 The equations representing the deformation under torsion of an elastic, solid bar of any cross-section are of the same form as those for the displacement of a soap film stretched over a hole in a flat plate, the hole being of the same shape as the section of the bar. The mathematical solution of these equations may be difficult, but it is easy to measure the displacement of the soap film hence by forming a soap film on a box, in the lid of which is a hole of the same shape as the bar, and measuring the contour lines of the film when pressure is applied inside the box, by means of a spherometer, the effect of torsional stress on bars of the most complicated section may be ascertained. 

Compression testing is also useful for evaluating the technological and end-use properties of milk fat and butter (Davis, 1937 Scott-Blair, 1938 Dolby, 1941a Mohr and Wellm, 1948). In this approach, a uniform stress is applied to the top and bottom of a sample (typically a cylinder or prism) placed between two flat plates. Because of the uniform stress field, it is easier to extract rheological information than in penetrometery. 

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