Finite cladding

As initial distribution corresponds to the linear mode (2.11) of the given waveguide, the deviation of T z) with respeet to unity may he eonsidered as a measure of the error in this method. The results presented in Fig.2 allow one to analyze the accuracy of the method depending on the type of finite-difference scheme (Crank-Nicholson" or Douglas" schemes have been applied) and on the method of simulation of conditions at the interface between the core and the cladding for both (2D-FT) and 2D problems. 

Refined finite-difference schemes were obtained by introducing corrections at the interface between the core and the cladding according to the method proposed in" (the value of the transverse grid step Ax was chosen to meet the condition d = NAx where N is an integer). 

The calculations in the preceding section have been performed for a simple planar slab waveguide of infinite extent but finite thickness sandwiched between semi-infinite substrate and cover layers. Other waveguide configurations are possible, most notably the cyhndricaUy symmetrical case of a high refractive index rod surrounded by a lower index cladding. This is the fiber-optic configuration used widely in telecommunications. 

TWINKLE is a multidimensional spatial neutron kinetics code, whieh is patterned after steady-state codes currently used for reactor core design. The code uses an implicit finite-difference method to solve the two-group transient neutron diffusion equations in one, two, and three dimensions. The code uses six delayed neutron groups and contains a detailed multi-region fuel-clad-coolant heat transfer model for calculating point-wise Doppler and moderator feedback effects. The code handles up to 2000 spatial points and performs its own steady-state initialisation. Aside from basic cross-section data and thermal-hydraulic parameters, the code accepts as input basic driving functions, such as inlet temperature, pressure, flow, boron concentration, control rod motion, and others. Various edits are provided (for example, channel-wise power, axial offset, enthalpy, volumetric surge, point-wise power, and fuel temperatures). 

Fig. 4.22 Belt-line reactor trip from full power. Time variation of hoop stress in vessel material from clad/vessel interface. 3DD three-dimensional finite-element analysis (Bangash) M. Marshal (From Bangash )...

These values of the invariants correspond to a class of rays called tunneling rays. The name tunneling is derived from the fact that these rays appear to tunnel a finite distance into the cladding [1], in analogy to the mechanism of frustrated total internal reflection [4], To deduce this, we note in Eq. (2-21) that g p) < 0 when Eq. (2-22) is satisfied. However, if we examine g r) in the... 

For the finite values of A for light, the evanescent fields lose some of their power to the absorbing cladding. This loss in turn leads to a loss of power from the ray path. Ray power flows along narrow ray tubes, as discussed at the beginning of Chapter 4, from which loss can occur only at reflection or turning points, i.e. at positions where there is an abrupt discontinuity in the tube or its cross-sectional area is zero. At these points, we express the loss in terms of a power transmission coefficient T, also known as a loss coefficient, defined by the dimensionless ratio... 

We recall from Chapters 1 and 2 that Snell s laws and the ray-path equation tell us that certain rays within the core of a waveguide will undergo refraction at the core-cladding interface. These are the reacting rays. The ray equation further tells us that other rays within the core of a fiber have an associated path in the cladding which extends indefinitely from some finite radial position beyond the core-cladding interface. These are the tunneling rays [1, 2]. In ... 

In this limit, all of a mode s power is concentrated in a finite region about the maximum core index, i.e. about n = n. For clad profiles this region is entirely within the core. 

The solution U for the fundamental mode (/ = 0, m = 1) is plotted against V in Fig. 14-8(a) for various values of q. As V increases, the value of U approaches a finite limit only for the step profile. For all other profiles U is unbounded as F-4 (X). If we compare U with the analytical form of Eq. (14-9) for the infinite power-law profiles when F = 8, and use the values of G(q) in Fig. 14-2(c), the error is less than 3.5 % for the values of q shown. In other words, most of the fundamental-mode power is confined within the core, where the clad and infinite power-law profiles are identical, and the region beyond R = 1 has little effect. 

Example Dipole within a step-profile fiber 25-15 Example Tubular source within a step-profile fiber 25-16 Effect of a finite cladding... 

Finite element techniques are the commonest numerical technique today for two- and three-dimensional elastic problems and they can also be applied to one-dimensional problems. The fuel and clad are divided into elements or annuli of finite thickness. Forces and displacements are defined for the inner and outer radii of each annulus and for the axial direction. Stress, strain, and temperature are assumed constant within each annulus and are related to the forces and displacements, respectively. For example, the strains in an element / are given by... 

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