Numerical cell model

As already discussed, modelling this multiple exponential decay function with the numerical cell model gives valuable information about cell morphology and membrane permeability. Similar information is available... 

The Ti q) behave as wavevector-dependent relaxation times and the form of the wavevector dependence can provide a useful check on the consistency of models. Table 5 shows a comparison of the experimental coefficients for fresh apple tissue with those calculated with the numerical cell model. The agreement is quite reasonable and supports the general theoretical framework. It would be interesting to apply this approach to mealy apple and to other types of fruit and vegetable. 

Very few of the references in Tables 1-3 attempt any quantitative modelling of their NMR data in terms of cell microstructure or composition. Such models would be extremely useful in choosing the optimum acquisition pulse sequences and for rationalising differences between sample batches, varieties and the effects of harvesting times and storage conditions. The Numerical Cell Model referred to earlier is a first step in this direction but more realistic cell morphologies could be tackled with finite element and Monte Carlo numerical methods. 

Most recently these ideas have been combined with a numerical cell model to relate S(q, A, r) to cell structure in plant parenchyma tissue.143 Using PGSE data for apple tissue a value for the plasmalemma membrane permeability was estimated. The application of this numerical cell model to mammalian tissue might enable quantitative interpretation of diffusion weighted contrast in clinical MRI. Table 6 lists a number of other applications of the PGSE method to food-related materials, although few of these studies have attempted to explore systematically the whole of the three-dimensional q—A—r space. 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

Numerous model calculations correlating aqueous VPIE s using simple harmonic or pseudo-harmonic cell models have been reported (see Fig. 5.8 and Table 5.8 for an ultra-simple version). Such calculations show the importance of the librational hydrogen bonded modes and the stretch-libration interaction in determining VPIE for D or T substitution. 

Wang, G., Mukherjee, P P, and Wang, C. Y. Optimization of polymer electrolyte fuel cell cathode catalyst layers via direct numerical simulation modeling. Electrochimica Acta 2007 52 6367-6377. 

Due to the complexity and interconnectivity of the governing equations and constitutive relationships, most fuel-cell models are solved numerically. Al-... 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

Section 2.1 gives a generalized summary of fuel cell models, while section 2.2 discusses the need for employing large numerical meshes and hence advanced numerical algorithms for efficient fuel cell simulations. Section 2.3 briefly reviews the efforts, in the literature, to measure basic materials and transport properties as input to fuel cell models. 

Numerical calculations for the residual stresses in the anode-supported cells are carried out using ABAQUS. After modeling the geometry of the cell of the electro-lyte/anode bi-layer, the residual thermal stresses at room temperature are calculated. The cell model is divided into 10 by 10 meshes in the in-plane direction and 20 submeshes in the out-plane direction. In the calculation, it is assumed that both the electrolyte and anode are constrained each other below 1400°C and that the origin of the residual stresses in the cell is only due to the mismatch of TEC between the electrolyte and anode. The model geometry is 50 mm x 50 mm x 2 mm. The mechanical properties and cell size used for the stress calculation are listed in Table 10.5. 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

This means that the numerical results can give but a rough and qualitative picture of some basic effects. In addition, the cell model turned out to possess several shortcomings, the most serious of which are the strong correlation between the number of cells per unit volume and the axial and radial flow characteristics and the fact that an excessive number of cells is needed to model a strong thermal runaway reaction with high temperature peaks. 

An analysis of radial flow, fixed bed reactor (RFBR) is carried out to determine the effects of radial flow maldistribution and flow direction. Analytical criteria for optimum operation is established via a singular perturbation approach. It is shown that at high conversion an ideal flow profile always results in a higher yield irrespective of the reaction mechanism while dependence of conversion on flow direction is second order. The analysis then concentrates on the improvement of radial profile. Asymptotic solutions are obtained for the flow equations. They offer an optimum design method well suited for industrial application. Finally, all asymptotic results are verified by a numerical experience in a more sophisticated heterogeneous, two-dimensional cell model. 

The numerical expressions of oty and depend on the adopted simplified cell model and, with the exception of the simplest cases, cannot be calculated a priori. For spherical cells the ratio... 

Lindstrom Carlsson, 1993 Rotmensch et ah, 1994). Spheroid-like hollow bodies with a multicellular epithelial morphology have also been utihzed as a model of pathogenesis of infection as a result of Neisseria infection. Numerous cell-cell contacts representative of cells in vivo have been demonstrated in spheroids from nasopharyngeal cells, e.g. junctional complexes, desmosomes, specific orientation of the cytoskeleton and cellular organelles (Boxberger et aL, 1993). 

Before presenting numerical results, it is worth summarizing the main characteristics of the experimental results for the osmotic pressure of polyelectrolyte solutions [9, 17, 18, 57, 107], The measured osmotic coefficients most often exhibit strong negative deviations from ideality. The measured values are a) lower than it was predicted by the cylindrical cell model theory, b) rather (but not completely) insensitive to the nature of the counterions, and c) also insensitive to the polyelectrolyte concentration in a dilute regime and/or for... 

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