TANGENTIAL

The simplest type of centrifugal device is the cyclone separator (Fig. 3.4), which consists of a vertical cylinder with a conical bottom. The centrifugal force is generated by the fluid motion. The mixture enters in a tangential inlet near the top, and the rotating motion so created develops centrifugal force which throws the particles radially toward the wall. 

Cyclones. Cyclones are also primarily used as prefilters. These also were discussed in Chap. 3 and illustrated in Fig. 3.4. The particle-laden gas enters tangentially and spins downward and inward, ultimately leaving the top of the unit. Particles are thrown radially outward to the wall by the centrifugal force and leave at the bottom. 

The wall thickness estimation in tangential projection technique is based on the evaluation of profile plots along the pipe diameter as shown in fig. 1 (lowest row). 

Fig. 1 Simulated, profiles across a pipe in the tangential projection technique and the results of different filters for determination of the projected wall thickness...

The detection of the profile edges gives the projected wall thickness in pixels of the image data. The next step of data processing is the compensation of the magnification factor in the used tangential projection method. 

To consider the geometrical magnification of the pipe image on film or IP in the tangential exposure technique according to the set-up shown in fig. 4 a correction of the measured wall thickness (w ) must be performed. The true wall thickness (w) depends in a rather complicated way on the film-focus-distance (f), the radius of the pipe (r) and the radius of the insulation (R) as shown in equation 2 ... 

Fig.4 Geometrical magnification effect inherent to the tangential exposure technique...

Fig.6 User interface for automated wall thickness estimation in tangential projection technique...

Projection radiography has long been used for pipe inspection and corrosion monitoring. In this traditional tangential wall thickness estimation the distance of border lines of the projected wall shadows of a pipe onto the film is a direct measure for the wall thickness. This method is not considered here, newer developments can be found in / /. 

Basically it is only possible to determine a wall thickness change in penetration direction from a density variation of a radiographic film. No absolute wall thickness values can be obtained in this way contrary to the tangential method. 

The moving force of a flow is the gradient of tangential component of the pressure p in a film, which is defined as the difference between the pressure pn in a bulk phase above a... 

The boundary conditions at the z=0 surface arise from the mechanical equilibrium, which implies that both the nonnal and tangential forces are balanced there. This leads to... 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

The normal component of velocity and tangential component of surface force are set to zero along a line of symmetry. For the domain shown in Figure 3.3 these are expressed as... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

Consider a solid wall section as is shown in Figure 3.4. The following relationships between the components of unit outward normal and tangential vectors are true at all points... 

Next, suppose we consider the tangential velocity v of segment i in a polymer molecule. The segment is located a distance r from the center of mass of the molecule and possesses an average angular velocity co. The situation is sketched in Fig. 2.12a. Since v = rco, it follows that the x and y components of the velocity are given by... 

Now we must consider how the frictional force experienced by this segment is related to the tangential velocity whose components are given by these expressions. 

Items (1) and (2) mean that the refractive index in the tangential direction of the spherulite is generally greater than that along the radius. 

Here t = (—n2,ni) is the tangential unit vector at T. Integrating by parts, one can obtain the Green formula (Temam, 1983 Khludnev, Sokolowski, 1997)... 

We continue the investigation of the contact problem for a plate under creep conditions. In this section the case of both normal and tangential displacements of the plate is considered. 

One can represent the vector aijPj on F as a sum of the normal and tangential components... 

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