Network of resistors

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

Figure 7.12 A network of resistors can be used to perform the last step of the algorithm, i.e. the integration. Each point of the grid is connected to its immediate neighbors by a resistor of resistance R. It is also connected to its diagonal neighbors by a resistor of resistance 4R.

Let us consider the one-dimensional case again. Figure 10.14 shows a linear network of resistors. If we carry out the same derivation as mentioned earlier but now for a onedimensional resistive grid, we arrive at the following equation (Jahne 2002) ... 

EXAMPLE 21.5. Still another example is finding the resistance of a network of resistors (Figure 21.3). 

The experimental data for a were compared with theoretical values calculated by means of analogy considerations to electric flow (Ohm s law) [112]. Simple circuit models based on a network of resistors were applied to simulate the cross-type configuration chosen. It could be shown that the calculated values for amin and am3X were in excellent agreement with the experimental data. 

If external potentials are applied to a system of several interconnected channels, the respective field strength in each channel will be determined by Kirchhoff s laws in analogy to an electrical network of resistors [28]. Ideally, electrokinetically driven mass transport in each of the channels will take place according to magnitude and direction of these fields. This allows for complex fluid manipulation operations in the femtoliter to nanoliter range without the need of any active control elements, such as external pumps or valves. This is of particular relevance due to the demanding limitations with respect to void volumes in the system (see Sect. 2). 

In deriving the theory below, we will rely frequently on analogies between the electrochemical cell and networks of resistors and capacitors that are thought to behave like the cell. This feature may seem at times to disembody the interpretation from the chemical system, so let us emphasize beforehand that the ideas and the mathematics used in the interpretation are basically simple. We will do our best to tie them to the chemistry at every possible point, and we hope readers will avoid letting the details of interpretation obscure their view of the great power and beauty of these methods. 

Figure 1.14. Schematic of steps that are part of an overall reaction from initial reactants to final products via intermediate species. indicates a reaction intermediate and the different -values indicate that there are many resistances for processes in parallel and in series. A chemical reaction is analogous to a network of resistors in circuits a chemical conversion rate is comparable to a current. The overall kinetics are determined by the lowest k-value in the reaction series that has the highest rate of production of end products compared to other reaction series running in parallel. The scheme presupposes a small driving force AG.

For typical catalyst layers impregnated with ionomer, sizes of hydrated ionomer domains that form during self-organization are of the order of 10 nm. The random distribution and tortuosity of ionomer domains and pores in catalyst layers require more complex approaches to account properly for bulk water transport and interfacial vaporization exchange. A useful approach for studying vaporization exchange in catalyst layers could be to exploit the analogy to electrical random resistor networks of... 

Figure 2 The calculated potential distribution across a DSSC modeled by the resistor network of Fig. 1. The node number corresponds to distance through the cell in this case, the cell was modeled with 100 circuit elements. See the text for details.

Immittance — In alternating current (AC) measurements, the term immittance denotes the electric -> impedance and/or the electric admittance of any network of passive and active elements such as the resistors, capacitors, inductors, constant phase elements, transistors, etc. In electrochemical impedance spectroscopy, which utilizes equivalent electrical circuits to simulate the frequency dependence of a given elec-trodic process or electrical double-layer charging, the immittance analysis is applied. 

Electric circuits or networks can be analyzed using both Ohm s law and KirchhofFs laws. For a circuit of resistors in series, as shown in Figure 2.6, the current flow in each resistor is the same (IR). 

A stored quantity can be produced, conserved, or dissipated. For example, the sum of kinetic and potential energy-is conserved in a mechanical system without friction. However, the presence of friction will dissipate some of the useful mechanical energy. Similarly, in a network of capacitors and resistors, only part of the supplied energy is stored in the capacitors while the rest is dissipated as heat in the resistors. 

It is possible to represent the entire electrochemical system including the instrumentation (potentiostat, etc.) as a single electrical circuit. The solution is usually spatially discretized into a network of resistance elements (see for example Coles et al., 1996). Double-layer charging can also be incorporated into these models by defining each element to contain a capacitor as well as a resistor. 

If we have a circuit (or network) constituted only of resistors, the voltage at any point in it is uniquely defined by the applied voltage. If the input varies, so does this voltage — instantly, and proportionally so. In other words, there is no lag (delay) or lead (advance) between the two. Time is not a consideration. However, when we include reactive components (capacitors and/or inductors) in any network, it becomes necessary to start looking at how the situation changes over time in response to an applied stimulus. This is called time domain analysis. ... 

Local-field effects are similarly important in conductivity problems. Consider a simple-cubic network of identical resistors from which a fraction 1 -p has been removed. The conductivity has the form... 

Cable theory Representation of a cylindrical fiber as two parallel rows of resistors (one each for the intracellular and extracellular spaces) connected in a ladder network by a parallel combination of resistors and capacitors (the cell membrane). 

In order to compute the thermal conductivity tensor kj, the multitude of walls shown in Fig. 3.7 can be regarded as a network of thermal resistors. On this basis, the components of the thermal conductivity tensor are derived as... 

The simulations of conductivity in random resistor-capacitor (RC) networks confirmed freqnency dependence of Eqnation (65) (Panteny et al., 2005). In random networks percolated resistors lead to the DC plateau at low frequencies. In contrast, non-percolated resistors generate negative deviation of conductivity from DC plateau and non-monotonons decrease of conductivity at low frequencies. These regular or irregular deviations from DC plateau are in polymer-salt systems due to electrode polarization. At high frequencies random network simulations recovered the power-law dependency of conductivity as in Equation (65). Generally, one may say conductivity in random RC networks will be preferably determined by resistors at low frequencies and by capacitors at high frequencies. 

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