Cylindrical coordinate system, model

A similar mathematical model to that just described for bench slot exhausts can again be used, but in this case the Laplace equation should be employed in a cylindrical coordinate system (see Fig. 10.83), namely,... 

A model that employs a two-dimensional cylindrical coordinate system and assumes axial symmetry with respect to r- and z-axes is developed. Figure 3.2.2 shows the coordinate system, computing region, and... 

The charge density of dust transported through ducts and the resultant electric fields at the duct Inner walls was monitored by a Monroe Electronics Inc., Model 171 electric fieldmeter. All the electrostatic sampling In the field was performed In circular cross-section ducts. Thus, the electrostatic field Intensity, for this geometry, can be determined from Poisson s equation using the cylindrical coordinate system. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

Let us consider a shallow fluidized bed combustor with multiple coal feeders which are used to reduce the lateral concentration gradient of coal (11). For simplicity, let us assume that the bed can be divided into N similar cylinders of radius R, each with a single feed point in the center. The assumption allows us to use the symmetrical properties of a cylindrical coordinate system and thus greatly reduce the difficulty of computation. The model proposed is based on the two phase theory of fluidization. Both diffusion and reaction resistances in combustion are considered, and the particle size distribution of coal is taken into account also. The assumptions of the model are (a) The bed consists of two phases, namely, the bubble and emulsion phases. The voidage of emulsion phase remains constant and is equal to that at incipient fluidization, and the flow of gas through the bed in excess of minimum fluidization passes through the bed in the form of bubbles (12). (b) The emulsion phase is well mixed in the axial... 

Let a CNT bundle contains N infinitely long metallic CNTs, closely packed together, with surface conductivity aQ. The bundle radius Rb is much less than the wavelength A. Since the incident field is almost homogeneous over the bundle cross-section, a symmetrical surface wave is excited in the bundle. In order to take into account the symmetrical local field distribution inside and outside the bundle we model one as a system of n coaxial thin-walled cylinders with the radii R, (l = l,2.,.n, Rb = Rn> Rn x> > Rx) and the surface conductivity cr, /(2 R,), where cr, is equal to the sum of linear conductivities of CNTs placed between the surfaces of cylinders with radii R, and R,, . Boundary conditions for electric Hertz potential on the surface of I -th cylinder in the cylindrical coordinate system (p, q>, z) is as follows [2] ... 

A new model has been developed, dealing with the current density and the consequent Joule heat distribution between the specimen and the die [25]. Thermal balances, as given in Eq. (6.2), where Joule heat is expressed in terms of voltage gradient, are coupled to the current density balances, i.e., Kirchhoff law with distributed parameters in a 2D cylindrical coordinate system ... 

The relevance of contact resistances in SPS process has been simulated and confirmed, with the simulation shown in Fig. 6.15 as an example [4]. The system simulated is a Model 1050-Sumitomo SPS, where a solid graphitic cylinder is inserted into the die. The 2D cylindrical coordinate system of coupled thermal and electrical problems is numerically solved by using Abaqus (FEM). The heat losses due to radiation from all exposed surfaces, except those on the ends of the rams, have been considered, where a constant temperature of 25 °C is used for the simulation. Thermophysical parameters of all materials are available in that study. A proportional feedback controller based on the outer surface temperature of the die is modeled, in order to determine the voltage drop applied at two ends of the rams. This controller is used to imitate the actual proportional integral derivative (PID), which is observed in real SPS facilities. It is used to apply electric power input to the system when experiments are conducted in terms of temperature controlling. 

The computer simulation of the experiments in the frame work of two-phase single-velocity model [7] has been carried out in order to clear up the mechanism leading to the formation and irreversible development of the cavitation zone. The one-dimensional problem was solved in the cylindrical coordinate system ( T, 0,2 ). The impulsive energy release occurs in time moment t=0 along the axis Z. When t > 0 three areas are singled out in the problem are the explosive products (0 < p < ),... 

Since we model the essential part of the event by a rod with an axisymmetric strain field the velocity field in the event can be given in canonical form for a cylindrical coordinate system as... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

Although the foregoing example in Sec. 4.2.1 is based on a linear coordinate system, the methods apply equally to other systems, represented by cylindrical and spherical coordinates. An example of diffusion in a spherical coordinate system is provided by simulation example BEAD. Here the only additional complication in the basic modelling approach is the need to describe the geometry of the system, in terms of the changing area for diffusional flow through the bead. 

As mentioned in the introduction to this chapter this is a necessary condition when approximating the cylindrical screw in the Cartesian coordinate system. The screw rotation theory, New Theory line, predicts that the rate should constantly increase as the channel gets deeper. When a fixed positive pressure occurs for the screw rotation model, the New Theory with Pressure line, the predictions fits the data very well for all H/Ws. Thus for modern screw designs with deeper channels, reduced energy dissipation, and lower discharge temperatures, the screw rotation model would be expected to provide a good first estimation of the performance of the extruder regardless of the channel depth for Newtonian polymers. 

It is not unusual to encounter a problem that is not conveniently posed in one of the common coordinate systems (i.e., cartesian, cylindrical, or spherical). As an illustration consider the flow behavior for the system shown in Fig. 5.20. The analysis seeks to understand the details of the flow field and pressure drop in the narrow conical gap between the movable flow obstruction and the conical tube wall. Intuitively one can anticipate that the flow may have a relatively simple behavior, with the flow parallel to the gap. However, such simplicity can only be realized when the flow is described in a coordinate system that aligns with the gap. An orthogonal curvilinear coordinate system can be developed to model this problem. 

Let us proceed to define appropriate coordinate systems (see Fig. 1 and Table 1). We use four different levels of coordinate frames in the averaging procedures of our model construction scheme to describe the molecular arrangement in the scattering volume. On the first level, System M describes the real structure of a segment, whereby the origin is fixed to a well-defined molecular unit System MC (second level) is defined in cylindrical coordinates with the symmetry axis given by the long axis of the rod-like molecular... 

From the extensive experimental and model development work performed at CSM (during a period of over 15 years), it has been demonstrated that a heat transfer controlled model is able to most accurately predict dissociation times (comparing to laboratory experiments) without any adjustable parameters. The current model (CSMPlug see Appendix B for details and examples) is based on Fourier s Law of heat transfer in cylindrical coordinates for the water, ice, and hydrate layers, and is able to predict data for single- and two-sided depressurization, as well as for thermal stimulation using electrical heating (Davies et al 2006). A heat transfer limited process is controlled by the rate of heat supplied to the system. Therefore, a measurable intermediate (cf. activated state) is not expected for heat transfer controlled dissociation (Gupta et al., 2006). 

Indeed, we made the choice of the coordinates system (cartesian, cylindrical or spherical) which will be used for our actual case and the model equation will be transformed for the selected coordinates system. 

Figure 6.14 Coordinate system used to model the diffusion domain for a cylindrically approximated diffusion domain. The plane to be simulated is shaded [40].

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