Distributed Element Model

Microwave in Microfiuidics, Fig. 3 Distributed element model for transmission lines. On a transmission line segment with a length of Ax, the electrical signal experiences a series inductance due to the magnetic field generated by the current, a series resistance due to the conductive losses over the conductive wire, a parallel capacitance due to the electric field developed between the two conductive lines, and a parallel conductance due to the current leaked through the insulating material... 

Further Empirical and Semiempirical Models. Although various empirical distributed-element models have already been discussed, particularly in Section 2.2.2.2, the subject is by no means exhausted. Here we briefly mention and discuss various old and new elements which may sometimes be of use in a fitting circuit such as that of Figure 2.2.10. Complex plane plots of IS data by no means always yield perfect or depressed semicircular arcs often the arc is unsymmetric and cannot be well approximated by the ZC. An unsymmetrical impedance plane arc usually exhibits a peak at low frequencies and CPE-like response at sufficiently high frequencies. The reverse behavior is not, however, unknown (Badwal [1984]). An expression originally proposed in the dielectric field by Davidson and Cole [1951] yields ordinary asymmetric behavior. Its h generalization is... 

Figure I Conceptual sketch for the Distributed Element Model (DEM) (taken from Chiang 1992).

Iwan, W.D. 1966. A distributed-element model for hysteresis and its steady-state dynamic response. Journal of Applied Mechanics, 33, 893-900. 

Finite element modelling of flow distribution in an extrusion die... 

Nassehi, V., Kinsella, M. and Mascia, 1.., 1993b. Finite element modelling of the stress distribution in polymer composites with coated fibre interlayers. J. Compos. Mater. 27, 195-214. 

Nassehi, V. and Pittman,. 1. F. T., 1989. Finite element modelling of flow distribution in an extrusion die. In Bush, A. W., Lewis, B.A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, Chapter 8, Ellis Horwood, Chichester. 

From the start, we knew we needed large anodes to meet the challenge of inexpensive fluorine these calculations clearly show the need for a better design for large anodes. The obvious solution is to put a metal conductor down the middle of the anode. Figure 20 shows the results from a finite-element model of the temperature distribution in such an improved large anode with a central metal conductor. 

In Section II,C we have deliberately chosen a simple set of problem specifications for our steady-state pipeline network formulation. The specification of the pressure at one vertex and a consistent set of inputs and outputs (satisfying the overall material balance) to the network seems intuitively reasonable. However, such a choice may not correspond to the engineering requirements in many applications. For instance, in analyzing an existing network we may wish to determine certain input and output flow rates from a knowledge of pressure distribution in the network, or to compute the parameters in the network element models on the basis of flow and pressure measurements. Clearly, the specified and the unknown variables will be different in these cases. For any pipeline network how many variables must be specified And what constitutes an admissible set of specifications in... 

Fan, C.F. and Hsu, S.L. (1992a) A study of stress distribution in model composites by using finite-element analysis. I. End effects. J. Polym. Sci. Part B. Polym. Phy. 30, 603-618. 

It is not clear what is the best model to describe the variability of soil metal concentrations. Ahrens (1954, 1966) has proposed that the distribution of elements in igneous rocks approximates to a log-normal distribution. This model does not necessarily apply to soils but the available evidence suggests it may. Its applicability underlies the interpretation of geochemical data in mineral exploration. 

Figure 6. Remanence enhancement in a two-phase Nd2Fe 4B/Fe3B magnet containing 343 grains. Left Finite element model of the grain structure. Right Magnetization distribution in a slice plane for zero applied field. The arrows denote the magnetization direction projected on a slice plane.

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

Figure 3.5 Equivalent-circuit models to describe the near-resonant electrical characteristics of the resonator (a) distributed model (b) lumped-element model. (Reprinted with permission. See Refs. [7 14J. (a) 1994 American Institute of Physics and (b) 1993 American Chemical Society.)...

The rheology determines a distribution of residence time in the barrel, and the resultant heat transfer characteristics. Even with simple flow behaviour, finite-element modelling predicts greater shear rates and heating at the walls. This explains the observations by Richmond, that homogeneous melt structures first form at the wall, and further implies that on exit from the die, the melt stream may not be homogeneous with regard to its shear and/or temperature history. 

A common application of this equation to trace element modeling is to examine the variations in trace element abundances and ratios for elements with different bulk distribution coefficients (Figure 7). In this plot, F is the fraction of melt for equilibrium crystallization, F proceeds from... 

Figure 8 Distribution of Be, B, Rb, Sr, Y, Ce, and Ba between minerals of average MORB and peUte in blueschist and eclogite facies (employing representative mineral modes for natural bluescbists and experimental epidote-eclogite, and trace element concentration data mostly from Domanik et al, 1993). At subsoUdus temperatures, diffusive equilibration is ineffective (except for micas) and tbe equiUbrating volume that needs to be taken into account for trace-element modeling is defined by tbe reacting minerals. Thus, a given trace element equilibrates with tbe fluid only when its host phase(s) break(s) down.

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

Ideally, first the measurement modeling should be carried out. The number and the nature of the circuit elements should be identified and then the process modeling should be carried out. Such a procedure is relatively elementary for a circuit containing simple elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation CPE, semi-infinite, finite length, or spherical diffusion, etc. However, many different conditions arise from the numerical calculations (e.g., for correct solution for porous electrodes, for... 

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