Finite differences model

If energy is supplied to or extracted from a layer within the component, finite-difference models or problem-adapted one-dimensional response-factor- based models have to be used. 

Finite difference models to simulate the diffusion and advection of oil in water have been developed and tested in wave basins [1748]. 

In a finite difference model, the differential equation representing mass transport (Eqn. 20.24 or 20.25) is converted into an approximate, algebraic form that can readily be evaluated using a computer. A derivative of concentration in space evaluated between nodal points (/, J) and (7 + 1, J), for example, can be written,... 

Lassiter, R. R., and D. W. Hayne. A finite difference model for simulation of dynamic processes in ecosystems, pp. 367-440. In B. C. Patton, Ed. Systems Analysis and Simulation In Ecology. Vol. 1. New York Academic Press. 1971. 

The creep stress was assumed to be shared between the polymer structure yield stress and the cell gas pressure. A finite difference model was used to model the gas loss rate, and thereby predict the creep curves. In this model the gas diffusion direction was assumed to be perpendicular to the line of action of the compressive stress, as the strain is uniform through the thickness, but the gas pressure varies from the side to the centre of the foam block. In a later variant of the model, the diffusion direction was taken to be parallel to the compressive stress axis. Figure 10 compares experimental creep curves with those predicted for an EVA foam of density 270 kg m used in nmning shoes (90), using the parameters ... 

This approach of subdividing space into an increasing number of discrete pieces provides the basis for many numerical computer models (e.g., the so-called finite difference models) an example will be discussed in Chapter 23. Although these models are extremely powerful and convenient for the analysis of field data, they often conceal the basic principles which are responsible for a given result. Therefore, in the next chapter we will discuss models which are not only continuous in time, but also continuous along one or several space axes. In this context continuous in space means that the concentrations are given not only as steadily varying functions in time [QY)], but also as functions in space [C,(r,x) or C,(t,x,y,z)]. Such models lead to partial differential equations. A prominent example is Fick s second law (Eq. 18-14). 

To model a complete stack, which may be constituted of more than 1000 cells, it is necessary to adopt a different approach. In this chapter a finite difference model is presented. Only energy equation and current conservation are solved. This allows one to examine possible improvements in the stack configuration design that can be achieved by taking advantage of the relation between temperature and elec-tronic/ionic resistivity, heat transfer and chemical reactions, etc. In addition, this model can be used for analyzing the effects of possible anomalies and performance degradation. 

The zone method is an effective way to model radiation heat transfer when geometries are sufficiently simple as in this case (Hottel and Sarofim, 1967). The system is divided into subsystems, the zones, which can be either surfaces (in the case of solids) or volumes (in the case of non-transparent gases). In the case of the tubular SOFC, the zones are the internal and external surfaces of the cell slices, the external surface of the tube elements, the fuel elements. Each zone is considered as characterized by a unique temperature. A zone model is particularly suitable for to use in a model like the finite difference model introduced in Section 7.4.1. 

The finite difference air pressure model was compared with the analytical solution for depths of 1.0 and 2.0 m (Figure 13). As can be seen, the model matches the analytical solution to subsurface air pressure very well, thus verifying the accuracy of the finite difference model. 

Figure 13. Simulations of air pressure at depths of 1.0 m and 2.0 m for the analytical solution and the finite difference model,...

Table 9 The complexity of the mass transport at various electrode geometries together with suggested methods for efficient finite difference modelling.

Martel, F., and Wunsch, C. (1993). The North Atlantic circulation in the early 1980s—An estimate from inversion oFa finite-difFerence model. J. Phys. Oceanogr. 23, 898—924. 

Cathles L. M. (1979) Predictive capabilities of a finite difference model of copper leaching in low grade industrial sulfide waste dumps. Math. Geol. 11, 175—191. 

Recently, computational fluid dynamics (CFD) models have been developed to guide the development of new BO designs [62-67]. Baker et al. developed a two-dimensional finite-difference model to solve the Navier-Stokes equation and to predict... 

Baker DA, Holte JE, and Patankar SV. Computationally two-dimensional finite-difference model for hoUow-fihre blood-gas exchange... 

In essence, the use of this criterion establishes agreement between a solution obtained by finite difference modeling and a solution obtained by the integral equation method. 

As with the hand conv or dryer sub-model, rather than incorporate the ftill equilibrium and finite difference models into the whole-plant model, a set of poljnomial fiinctions was derived from the models which adequately represent the processes over the range of interest. Capital cost was based on a published correlation by Bridgwater [6], updated to 1998 prices ... 

The model described here has been developed from a metallurgical standpoint, rather than an electrochemical one and originated as a cellular automata (CA) finite difference model. This approach dealt with the evolution of a representative concentration and electrical potential throughout the electrolyte only [1], The model was able to predict morphological features such as localized corrosion pits and capping but was limited to qualitative simulation. However the CA method has found alternative applications in the growth of corrosion pits... 

VLEACH Unsaturated zone VLEACH is a one-dimensional, finite difference model developed to simulate the transport of contaminants displaying linear partitioning behavior through the vadose zone to the water table by aqeuous advection and diffusion. 

On closer inspection, relatively more accurate values for the tempemture peak and mean degree of cure have been found using the finite-element model, which has provided an overestimated value of the standard degree of cure. The finite-difference model seems to provide underestimated values for all parameters considered however, both analyses produce good agreement with experimental values. 

Wan and Isayev (1996) examined a hybrid approach of control-volume finite-element and finite-difference modelling of injection moulding of rubber compounds. The effect of vulcanization on viscosity and yield stress during cavity filling is reported. On comparing two versions of the modified Cross viscosity models - with and without the effect of cure - the use of a viscosity model that accounts for the cure was found to improve the accuracy of the cavity-pressure-prediction models. When the modified Cross model was further extended to include the yield stress and was implemented in the simulation program a significant improvement in the prediction of cavity pressure was obtained in the case of low injection speed. 

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations. Singapore World Scientific, 1994. 

Rule 6 For differential equations having N > 3 terms, it is effective to develop finite-difference schemes for diverse subequations comprising M < N terms, and then combine all schemes in a comprehensive finite-difference model. 

Figure 9.1. Simulated contours of present-day hydraulic heads (in feet) generated from the calibrated two-dimensional finite difference flow model developed by GeoTrans (1987b). Arrows indicate the directions of groundwater flow. Also shown are the finite difference model and the confined/unconfined aquifer boundaries. Star symbols indicate the two water samples used in inverse modeling.

Delgado, A.E. and Sun, D.-W. 2003. One-dimensional finite difference modelling of heat and mass transfer during thawing of cooked cured meat. Journal of Food Engineering 57(4) 383-389. 

The first accounted only for C02 retention caused by dilution to dissolved inorganic carbon. The second accounted for additional dilution to an adsorbed C phase predicted from C02 -loss experiments. The geochemical models were separately coupled with a two-dimensional, finite-difference model for gas diffusion to simulate the distribution of P C02 in the unsaturated zone near a disposal trench at a low-level radioactive waste-disposal site near Sheffield, Illinois. Comparison of simulated P C02 distribution with onsite data supported the presence of the adsorbed C phase. 

The simulations were performed using a finite-difference model for solution of the diffusion equation in porous media (. For modeling purposes, the cross section (Figure 1) was divided into a 3-layered, 36 by 53 block-centered grid. The layers represented the Toulon Member, the Radnor Till Member, and eolian silts (Roxana Silt and Peoria Loess). Physical properties of the unsaturated zone that were used for modeling were identical to those listed in Table I. 

In the next step, a facies type has to be assigned to each finite-difference model cell within the simulation domain. For this task, the experimental histogram of the clusters (clusterl - clusterM) and the cluster variogram models are used to generate conditioned equiprobable three-dimensional realizations of the facies fields described by categorical variables. For the facies-based approach used here, the three-dimensional conditional sequential indicator simulation method (SIS) for categorical variables... 

Pollock, D. W. (1988) Semianalytical computation of path lines for finite difference models. Ground Water 26(6), 743-750. 

Modeling In this case, the matrices describing the system can be determined based on a model of the machine tool structure, which can be expressed as a numerical finite element model (FEM) or a finite-difference model (FDM). 

The 1-D case is by far the best handled. Lee and Harleman (55) developed a useful finite-difference model. Later work by the MIT group with real-time, 1-D, unsteady flows has developed a finite element model which is very versatile, dealing with salinity intrusion (57, 91), general water-quality problems 18), temperature (7), and nitrogen dynamics 65), Mathematical models and users manuals are available for these cases. 

Several finite difference models have been presented. From the transport standpoint, the best two appear to be the stream-tube models by Holly 43) and... 

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