Mathematical modeling electrical

If the dynamic behaviour of a physical system can be represented by an equation, or a set of equations, this is referred to as the mathematical model of the system. Such models can be constructed from knowledge of the physical characteristics of the system, i.e. mass for a mechanical system or resistance for an electrical system. Alternatively, a mathematical model may be determined by experimentation, by measuring how the system output responds to known inputs. 

Chapters 15 through 17 are devoted to mathematical modeling of particular systems, namely colloidal suspensions, fluids in contact with semi-permeable membranes, and electrical double layers. Finally, Chapter 18 summarizes recent studies on crystal growth process. 

In 1821 Michael Faraday sent Ampere details of his memoir on rotary effects, provoking Ampere to consider why linear conductors tended to follow circular paths. Ampere built a device where a conductor rotated around a permanent magnet, and in 1822 used electric currents to make a bar magnet spin. Ampere spent the years from 1821 to 1825 investigating the relationship between the phenomena and devising a mathematical model, publishing his results in 1827. Ampere described the laws of action of electric currents and presented a mathematical formula for the force between two currents. However, not everyone accepted the electrodynamic molecule theory for the electrodynamic molecule. Faraday felt there was no evidence for Ampere s assumptions and even in France the electrodynamic molecule was viewed with skepticism. It was accepted, however, by Wilhelm Weber and became the basis of his theory of electromagnetism. 

Chalermchat, Y, Fincan, M., and Dejmek, P, Pulsed electric field treatment for solid-hquid extraction of red beetroot pigment mathematical modelling of mass transfer, J. Food Eng., 64, 229, 2004. 

Berggren, K.-F., and A.F. Sadreev. Chaos in quantum billiards and similarities with pure-tone random models in acoustics, microwave cavities and electric networks. Mathematical modelling in physics, engineering and cognitive sciences. Proc. of the conf. Mathematical Modelling of Wave Phenomena , 7 229, 2002. 

The variation of the electric potential in the electric double layer with the distance from the charged surface is depicted in Figure 6.2. The potential at the surface ( /o) linearly decreases in the Stem layer to the value of the zeta potential (0- This is the electric potential at the plane of shear between the Stern layer (and that part of the double layer occupied by the molecules of solvent associated with the adsorbed ions) and the diffuse part of the double layer. The zeta potential decays exponentially from to zero with the distance from the plane of shear between the Stern layer and the diffuse part of the double layer. The location of the plane of shear a small distance further out from the surface than the Stem plane renders the zeta potential marginally smaller in magnitude than the potential at the Stem plane ( /5). However, in order to simplify the mathematical models describing the electric double layer, it is customary to assume the identity of (ti/j) and The bulk experimental evidence indicates that errors introduced through this approximation are usually small. 

Once the ions have entered the skimmer tip, it is necessary to extract and focus them into the analyser by subjecting the charged ions to constant electric fields. In order to construct an elfective ion optical array, it is necessary to calculate the path followed by the ions in the electrostatic fields. We can resort to a number of mathematical models, such as SIMION, for a better understanding and optimisation of the ion-optical design for ICP-MS and the processes involved [12]. 

In order to study cathode flooding in small fuel cells for portable applications operated at ambient conditions, Tuber et al.81 designed a transparent cell that was only operated at low current densities and at room temperature. The experimental data was then used to confirm a mathematical model of a similar cell. Fig. 4 describes the schematic top and side view of this transparent fuel cell. The setup was placed between a base and a transparent cover plate. While the anodic base plate was fabricated of stainless steel, the cover plate was made up of plexiglass. A rib of stainless steel was inserted into a slot in the cover plate to obtain the necessary electrical connection. It was observed that clogging of flow channels by liquid water was a major cause for low cell performance. When the fuel cell operated at room temperature during startup and outdoor operation, a hydrophilic carbon paper turned out to be more effective compared with a hydrophobic one.81... 

The mathematical model is based on the conservation of mass, momentum, electrical charge, and energy, coupled with appropriate constitutive laws. 

The control of a fed-batch alcoholic fermentation process can be obtained by controlling the substrate concentration in the medium by manipulation of the feed flow. The fermentation process presents complicated kinetic mechanisms. In addition, there is the absence of accurate and reliable mathematical models as well as the difficulty of obtaining direct measurements of the process variables owing to a lack of appropriate on-line analyzers and sensors. Control systems are formed by a set of instruments and control mechanisms connected through electrical signals in the... 

Mathematical modeling and computer simulation have been applied for various flow studies in rectangular microchannels (see Table 3.1). An equation to describe the flow in a rectangular channel has been given [124]. Simulation of fluid flow can be conducted by solving the coupled Poisson and Navier-Stokes equation for fluid velocity [532]. However, this complicated computation has been simplified by solving the Laplace equation for the electric field because it is proportional to fluid velocity [321]. 

These results led us to analyze the relationship between carrier-wave frequency and power density. We developed a mathematical model (6) which takes into account the changes in complex permittivity of brain tissue with frequency. This model predicted that a given electric-field intensity within a brain-tissue sample occurred at different exposure levels for 50-, 147-, and 450-MHz radiation. Using the calculated electric-field intensities in the sample as the independent variable, the model demonstrated that the RF-induced calcium-ion efflux results at one carrier frequency corresponded to those at the other frequencies for both positive and negative findings. In this paper, we present two additional experiments using 147-MHz radiation which further test both negative and positive predictions of this model. 

Figure 2 displays the percent difference in mean efflux between the exposed and sham groups at different power densities of 50- and 147-MHz radiation, as well as lines connecting those power densities which would produce the same value of internal electric field intensity in the samples. These results support the mathematical model and demonstrate its usefulness in defining effective power densities over a range of carrier frequencies. 

Verbrugge M, Liu P. Microstructural analysis and mathematical modeling of electric double-layer supercapacitors. Journal of the Electrochemical Society 2005 152(5) D79-D87. 

The 1952 Hodgkin-Huxley model for membrane electrical potential is perhaps the oldest and the best known cellular kinetic model that exhibits temporal oscillations. The phenomenon of the nerve action potential, also known as excitability, has grown into a large interdisciplinary area between biophysics and neurophysiology, with quite sophisticated mathematical modeling. See [103] for a recent treatise. 

One of the characteristics of basic devices is that they cannot be split up into parts. A basic device can also be a signal transformer (a function tvhich transforms the input into output, such as the thermocouple that transforms the input temperature into an electrical tension). The process phases are connected and characterized quantitatively, from the vievrpoint of characteristic relations (equations), as in Fig. 1.4 [1.11-1.13]. This structured mathematical modelling development corre-... 

A general mathematical model for simulating particulate removal in gas-solid fluidized beds is presented. Model predictions of the fluidized bed filtration efficiencies, which include the possibility of electrical effects, are shown to compare well to the experimental results of various investigators. 

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