Axial directions

Let us assume that stress gradient in axial direction is present but smooth. Then we can use a perturbation method and expand the solution of equation (30) in a series. The first term of this expansion will be a solution of the plane strain problem and potential N will be equal to zero. The next terms of the stress components will contain potential N also. 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

Thus p is independent of r and v is axially directed, with magnitude... 

The mechanism for formation of the 3 methyl glycoside is shown The mechanism for for mation of the a isomer is the same except that methanol approaches the carbocation from the axial direction... 

Fig. 2. (a) Closed-end condition (b) open-end condition (4). The stress in the axial direction of a cylinder sealed under closed-end conditions is given by... 

CompoundShrinka.g e. In its simplest form (Fig. 8a) compound shrinkage consists of machining the inner radius of an outer component I, (Qp so that it is smaller than the outer radius of an inner component II, The difference between the two is known as the radial interference 5. To assemble the cylinders, outer component I is heated and/or inner component II cooled so that the outer component can be sHpped over the inner as shown in Figure 8b. When the temperature of the assembly returns to ambient, a compressive stress (pressure) is generated across the interface which simultaneously compresses the inner and expands the outer component and, in so doing, displaces radius (r/j by Uj and radius ( jj by U, Unfortunately, it is difficult to carry out this operation without setting up stresses in the axial direction (32). 

Thermal Stresses. When the wak of a cylindrical pressure vessel is subjected to a temperature gradient, every part expands in accordance with the thermal coefficient of linear expansion of the steel. Those parts of the cylinder at a lower temperature resist the expansion of those parts at a higher temperature, so setting up thermal stresses. To estimate the transient thermal stresses which arise during start-up or shutdown of continuous processes or as a result of process intermptions, it is necessary to know the temperature across the wak thickness as a function of radius and time. Techniques for evaluating transient thermal stresses are available (59) but here only steady-state thermal stresses are considered. The steady-state thermal stresses in the radial, tangential, and axial directions at a point sufficiently far away from the ends of the cylinder for there to be no end effects are as fokows ... 

There are four basic variations of the linear MHD channel (Fig. 5) which differ primarily in their method of electrical loading. The simplest is the two-terrninal Faraday or continuous electrode generator, Figure 5a, where a single pair of current-collecting electrodes spans the channel in the axial direction, short-circuiting the channel from end to end. Hence, for this configuration, = 0, andj can be obtained from equations 21 and 22 ... 

The mathematical model chosen for this analysis is that of a cylinder rotating about its axis (Fig. 2). Suitable end caps are assumed. The Hquid phase is introduced continuously at one end so that its angular velocity is identical everywhere with that of the cylinder. The dow is assumed to be uniform in the axial direction, forming a layer bound outwardly by the cylinder and inwardly by a free air—Hquid surface. Initially the continuous Hquid phase contains uniformly distributed spherical particles of a given size. The concentration of these particles is sufftcientiy low that thein interaction during sedimentation is neglected. 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-sphnes for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

In another land of ideal flow reactor, all portions of the feed stream have the same residence time that is, there is no mixing in the axial direction but complete mixing radially. It is called a.plugflow reactor (PFR), or a tubular flow reactor (TFR), because this flow pattern is characteristic of tubes and pipes. As the reaction proceeds, the concentration falls off with distance. 

Packed Red Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of Modeling of Chemical Reactions in Sec. 23 is an apphcation of such equations. 

The so-called propeUer-type fans with blades that are aerodynami-caUy designed (as seen in Fig. 10-68) can consist of two or more stages. The air in these fans enters in an axial direction and leaves in an axial direction. The fans usually have inlet guide vanes followed by a rotating blade, followed by a stationary (stator) blade. 

Centraugal Blowers These blowers have air or gases entering in the axial direction and being discharged in the radial direction. These blowers have 3 tvpes of blades, radial or straight blades, forward curved blades, and backward cui ved blades (Figs. 10-69-10-71). 

Centrifugal Compressors The flow in a centrifugal compressor enters the impeller in an axial direction and exits in a radial direction. 

The material balance with bulk flow in the axial direction z and diffusion in the radial direction rwith diffusivity D gives rise to the equation... 

In towers with inert packing, both radial and axial gradients occur, although conduction in the axial direction often is neglected in view of the preponderant transfer of sensible enthalpy in a flow system. 

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