Rectangular Cross-section

There is no analytical solution for the Poiseuille flow problem with a rectangular cross-section. It is only possible to find a Fourier sum representing the solution. 

The governing equation for flow inside rectangular cross-section channel is given by 

The boundary condition at the wall from the no-slip boundary conditions are 

Let us propose a trial solution for velocity distribution, u(y, z) as a double sine series given by 

Note that this trial solution automatically satisfies all the above-mentioned no-slip boundary conditions. 

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We consider a co-extrusion die consisting of an outer circular distribution channel of rectangular cross-section, connected to an extrusion slot, which is a slowly tapering narrow passage between two flat, non-parallel plates. The polymer melt is fed through an inlet into the distribution channel and flows into... 

Through the use of sequential electric (electrostatic) and magnetic fields (sectors) and various correcting lenses, the ion beam leaving the ion source can be adjusted so that it arrives at the collector in focus and with a rectangular cross-section aligned with the collector slits. For the use of crossed electromagnetic fields. Chapter 25 ( Quadrupole Ion Optics ) should be consulted. 

Both microwave and millimetre wave radiation can be channelled in any direction by a waveguide made from metal tubing of rectangular cross-section, the dimensions depending on the frequency range. The absorption cell is also made from waveguide tubing. 

Gut Rubber. To produce cut mbber thread, smoked mbber sheet or crepe mbber is milled with vulcanizing agents, stabilizers, and pigments. This milled stock is calendered into sheets 0.3—1.3 mm thickness, depending on the final size of the mbber thread desired. Multiple sheets are layered, heat-treated to vulcanize, then sHt into threads for textile uses (Fig. 2). Individual threads have either square or rectangular cross-sections. 

The actual flotation phenomenon occurs in flotation cells usually arranged in batteries (12) and in industrial plants and individual cells can be any size from a few to 30 m in volume. Column cells have become popular, particularly in the separation of very fine particles in the minerals industry and coUoidal precipitates in environmental appHcations. Such cells can vary from 3 to 9 m in height and have circular or rectangular cross sections of 0.3 to 1.5 m wide. They essentially simulate a number of conventional cells stacked up on top of one another (Fig. 3). Microbubble flotation is a variant of column flotation, where gas bubbles are consistently in the range of 10—50 p.m. 

In the soldedess wrap (Fig. 3) or wire-wrap connection, a wire conductor is coiled around the back end of the separable contact, which has a square or rectangular cross section (4). The corners of the soldedess wrap post and the areas of the wire that are in contact with it are severely deformed. In a propedy made wrap, the force required to sHde the wire along the post exceeds the breaking strength of the wire. The method is suitable only for soHd wire, and special tools are used to make this connection. 

Embedded. Rectangular-cross-section aluminum fin which is wrapped under tension and mechanically embedded in a groove 0.25 0.05 mm (0.010 0.002 in) deep, spirally cut into the outside surface of a tube. 

Sampie Specifications 15 in. (38.1 cm) diameter aluminum ring the ring has a rectangular cross section measuring 1 x 2 in. (214 x 5 cm) the ring is fitted with a series of brass inlet and outlet nozzles... 

Fig. 6-14 (above) Anode shapes for the internal protection of tanks, (a) Semicircular cross-section, (b) rectangular cross-section, (c) trapezoid cross-section. 

In heat transfer applications, this jacket is considered a helical coil if certain factors are used for calculating outside film coefficients. The equivalent heat transfer diameter, D, for a rectangular cross-section is equal to 4 w (w being the width of the annular space). Velocities are calculated from the actual cross-section of the flow area, pw (p being die pitch of die spiral baffle), and die effective mass flowrate W dirough die passage. The effective mass flowrate is approximately 60% of die total mass flowrate of die jacket. 

Polyethylene is injected into a mould at a temperature of 170°C and a pressure of 100 MN/m. If the mould cavity has the form of a long channel with a rectangular cross-section 6 mm X 1 mm deep, estimate the length of the flow path after 1 second. The flow may be assumed to be isothermal and over the range of shear rates experienced (10 -10 s ) the material may be considered to be a power law fluid. 

Using the data in Tables 5.1 and 5.2, calculate the flow lengths which would be expected if the following materials were injected at 100 MN/m into a wide rectangular cross-section channel, 1 mm deep. 

These rooms are usually rectangular with doors on one of the short sides for entrance and exit of materials and operators. Room sizes vary from 18 m to more than 300 m and the shapes range from cubic to elongated with rectangular cross-section. 

In the case of a rectangular cross-section, a variety of methods and corresponding measurement point locations exist." - Table 12.8 shows the required measuring points for the log-Tchebycheff rule, where the velocity distribution in the wall-connected elements is logarithmic and in the central elements polynomial. 

FIGURE 12.23 An example of measuring point locations for a rectangular cross-section. 

Most comparisons of a failure criterion with failure data will be for the glass-epoxy data shown in Figure 2-36 as a function of off-axis angle 0 for both tension and compression loading [2-21]. The tension data are denoted by solid circles, and the compression data by solid squares. The tension data were obtained by use of dog-bone-shaped specimens, whereas the compression data were obtained by use of specimens with uniform rectangular cross sections. The shear strength for this glass-epoxy is 8 ksi (55 MPa) instead of the 6 ksi (41 MPa) in Table 2-3. 

The elasticity approaches depend to a great extent on the specific geometry of the composite material as well as on the characteristics of the fibers and the matrix. The fibers can be hollow or solid, but are usually circular in cross section, although rectangular-cross-section fibers are not uncommon. In addition, fibeie rejjsuallyjsotropic, but can have more complex material behavior, e.g., graphite fibers are transversely isotropic. 

Figure 3-34 Halpin-Tsai Calculations (Circles) versus Foye s Calculations for E2 of Rectangular Cross-Section Fibers in a Diamond Array (After Hatpin and Tsai [3-17])...

The distribution of x z isotropic beam of rectangular cross section comes from integration of the the stress-equilibrium equation... 

However, the foregoing derivation is valid only for isotropic beams of rectangular cross section. For beams of nonrectangular cross section, the parabolic stress distribution is not correct. Also, for laminated beams, the parabolic distribution is most assuredly incorrect because of layer inhomogeneity. In fact, for laminated beams, we must expect different shapes of stress distribution in each layer as seen in Figure 6-19 for wide beams (there interpreted as cylindrical bending of a long strip, i.e., a special plate). 

M. J. E. Golay, Theory of cliromatography in open and coated tubular columns with round and rectangular cross-section, in Gas Chromatography Amsterdam 1958 (Amsterdam Symposium), Desty D. H. (Ed.), Butterwoiths Scientific Publications, London, pp. 36-55 (1958). 

Assuming that the beam in Figure 2-29 has a rectangular cross-section with a height of 1 ft and a width of 3 in. and given that L = 10 ft, a = 4 ft, and P = 1,000 lb, what are the maximum values of the shear and tensile stresses within the beam ... 

Shear Stress. For a rectangular cross-section, the maximum value of Q occurs at the neutral axis, and, because the width b of the beam is a constant 3 in., the maximum value of the shear stress occurs at the neutral axis. 

Derrick A semipermanent structure of square or rectangular cross-section having members that are latticed or trussed on all four sides. This unit must be assembled in the vertical or operation position, as it includes no erection mechanism. It may or may not be guyed. 

Fig. 4-16 Basic snap fit design for a cantilever beam with a rectangular cross-section.

A liquid is in steady state flow in an open trough of rectangular cross-section inclined at an angle f) to the horizontal. On what variables would you expect the mass flow per unit time to depend Obtain the dimensionless groups which are applicable to this problem. 

The data on critical Reynolds numbers in micro-channels of circular and rectangular cross-section are presented in Tables 3.5 and 3.6, respectively. We also list geometrical characteristics of the micro-channels and the methods used for determination of the critical Reynolds number. 

Ishii (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase regimes. AML Report ANL-77-47 Ide H, Matsumura H, Tanaka Y, Fukano T (1997) Flow patterns and frictional pressure drop in gas-liquid two-phase flow in vertical capUlary channels with rectangular cross section, Trans JSME Ser B 63 452-160... 

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