Single Plane Systems

The maximum stress in the bend is found by multiplying the stress by the stress intensification factor j3. 

Expansion stress, must be less than or equal to the allowable stress.  

Note It is recommended that should be taken from ANSI B31-3—Refinery Piping Code. 

The maximum stress will occur at the bend point Bamax = 1,120 (1.6) = 1,790 psi. 

Example 2 Given Same configuration as Example 1, carbon steel pipe operating at 640° F. Pipe size, D = 12-in. 

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Figure 7-32. Nomograph to solve single plane systems.

Bucket Elevators. In a bucket elevator, a series of buckets attached to an endless belt or chain are filled with material and lifted vertically to a head pulley or sprocket, where the material is dumped. The buckets are then returned back down to a tail pulley or sprocket at the bottom. Bucket elevators are not self-feeding. They must be fed at a controlled rate to avoid overfilling the buckets and damagiag the machinery. In the usual arrangement of a bucket elevator, the chain or belt path is vertical or steeply inclined ia a single plane. Special chain supported bucket systems that can travel ia two and three planes have been developed. 

The definitions and relationships of mass, stiffness, and damping in the preceding section assumed a single-degree of freedom. In other words, movement was limited to a single plane. Therefore, the formulas are applicable for all single degree of freedom mechanical systems. 

In systems of the type under consideration the aromatic rings lie in a single plane. 

For the sake of simplicity, we consider an example of a one-dimensional periodic system of length L with N atoms with one core electronic state per atom. The interatom space is a. The pseudo-valence electron is assumed to be in a single plane wave... 

For an estimate of the ultimate shear strength, r0, of a single domain based on the lattice parameters we use a simple shear plane system proposed by Frenkel [19]. As shown in Fig. 19 it consists of a linear array of periodic force centres resembling the polymer chain. According to this model the relation between the relative displacement x along the shear direction and the shear stress is given by... 

This expression, however, is only justified for 2D systems, where the particles are represented essentially by disks, which are confined in a single plane and the particle-particle contact occurs along a line, as shown in Fig. 13. So, the tangential component of the relative velocity is always in the same plane and no coordinate transformation is required. 

Figure 2.31. Schematic representation of the P/T equilibria in a simple two-component system (forming continuous solid and liquid solutions). In (a) a perspective view of the P-T-X diagram is shown in (b) its projection on the P/T plane. Notice the two single-component systems represented, for instance, for the component B by the three lines SB/G (sublimation line of B representing the gas/so lid equilibrium), SB/LB (melting equilibrium of B) and the boiling line LB/G. The solid solution is indicated by a. Notice in (a) the isobaric and isothermal sections of the diagrams (compare with Fig. 2.1).

Multi-row detector systems are referred to as cone-beam systems. With a moving conveyor they become helical cone-beam systems. The cone-beam designation is in contrast to the fan-beam geometry used in Figures 3 and 4, where the source and detectors are aU in a single plane. 

Phase rule of Defay-Crisp describing the number of degrees of freedom for a system having one single plane surface (monolayer) Multi-component monolayers consisting of immiscible amphiphiles exhibit the same surface pressure for phase transitions and collapse points as the corresponding one-component monolayers, while these surface-pressures are different for mixtures of miscible amphiphiles. 

The Stress-Range Concept. The solution of the problem of the rigid system is based on the linear relationship between stress and strain. This relationship allows the superposition of the effects of many individual forces and moments. If the relationship between stress and strain is nonlinear, an elementary problem, such as a single-plane two-member system, can be solved but only with considerable difficulty. Most practical piping systems do, in fact, have stresses that are initially in the nonlinear range. Using linear analysis in an apparendy nonlinear problem is justified by the stress-range concept... 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

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