Model-upscaling

Except for exploring its catalytic potential, CHMO from Acinetobacter has also been used as a model system for upscaling BVMO-mediated biocatalysis. 

Another approach consisting of the Liapounov-Schmidt reduction coupled with a perturbation argument is developed in the articles Balakotaiah and Chang (2003), Balakotaiah (2004) and Chakraborty and Balakotaiah (2005). It allows developing multi-mode hyperbolic upscaled models. 

Balakotaiah and Chakraborty introduce a four-mode hyperbolic model but with non-linear reactions, in three-dimensional geometry and with much bigger Damkohler number (Balakotaiah, 2004 Chakraborty and Balakotaiah, 2005). The effective model cannot be directly compared with our system (12)-(13). Nevertheless, in Section 3.1 we derive a four-mode hyperbolic model, analogous to the models from Balakotaiah (2004) and Chakraborty and Balakotaiah (2005). We show that it is formally equivalent to our model at the order This shows the relationship between the upscaled models... 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

We solve the same equations as in Section 4.1.1. Since a is very close to the threshold value a = 2, the difference between the solution to the effective equation obtained by taking the simple mean, at one side, and the solutions to the original problem and to our upscaled equation, are spectacular. Our model approximates fairly well with the physical solution even without adding the correctors (Table 4). Parameters are given at Table 3. 

Abstract In this paper we discuss a pore scale model for crystal precipitation and dissolution in porous media. We consider weak solutions in general domains and dissol-ution/precipitation fronts in thin strips. The latter yields an upscaled transport-reaction model. 

Defining the cross-section average of u as u = Jf u(t, x, z)dz and letting e 0, the triple u,v,w converges to the unique weak solution triple U, V, W of the upscaled one-dimensional transport-reaction model introduced in [3],... 

Upscaling in environmental risk assessment concerns, amongst other things, the extrapolation of ecotoxicological data derived from laboratory tests and model... 

As temporal upscaling methods for acceleration of KMC simulation become mature and more robust, I expect that they will have a significant impact on the modeling of biological reaction networks. 

Dunbabin, V. M., McDermott, S., and Bengough, A. G. (2006). Upscaling from rhizosphere to whole root system modeling the effects of phospholipid surfactants on water and nutrient uptake. Plant Soil 283, 57-72. 

To set process conditions successfully in industrial design, it is useful to first determine the optimal conditions and design on a model scale. When applied properly, upscaling and downscaling have immense advantages. One of the important, but often ignored, functions of small-scale research is to learn from blunders. ... 

The pore-scale model provides the basis for development of a statistical framework for upscaling from a single pore to a sample of variably saturated porous medium. The statistical distribution of pore sizes is modeled as a gamma distribution with the expected values of liquid configuration in pore space calculated from geometrical and chemical potential considerations within the statistical framework. Model predictions compare favorably with measured retention data, yielding similarly close fits to data as the widely used van Genuchten parametric model. Liquid-vapor interfacial area as a function of chemical potential is readily calculated from estimated retention parameters. Comparisons of calculated inter-... 

The review is organized as follows first, we discuss aspects of the unitary approach for combining adsorptive and capillary contributions, and present the new pore scale model of Tuller et al. (1999). The upscaling scheme of Or and Tuller (1999) for representing sample scale retention properties will be presented, followed by illustrative examples with measured characteristic data and a discussion of critical soil parameters. The role of liquid-vapor interfacial area will be highlighted by comparisons of model predictions with limited measurements. Finally, we introduce hydrodynamic considerations of unsaturated flow in films and comers leading to prediction of hydraulic conductivity of rough rock surfaces and unsaturated porous media. 

The upscaling scheme was applied lo Ihe pore scale model based on the SYL formulation, leading to Ihe development of closed-form expressions for sample scale... 

In an attempt to derive closed-form expressions for unsaturated hydraulic conductivity, we were forced to fix the shape of the gamma distribution by using a constant distribution parameter = 2. This led to a reduced flexibility, especially for soils with narrow pore size distribution. In Fig. 1-15 we show improved predictions of AXp) when the distribution parameter , is left as a free parameter (using a numerical scheme for the upscaling). The dashed line in Fig. 1—15b represents the numerically evaluated relative saturation curve ( , = 6) that is almost indistinguishable from the VG-Mualem model. A summary of resulting parameters is listed in Table 1-6. 

The pore scale model and the associated unitary approach were upscaled to represent a sample of partially saturated porous medium. 

Or, D., and M. Tuller. 1999. Liquid retention and interfacial area in variably saturated porous media Upscaling from single-pore to sample-scale model. Water Resour. Res. 35 3591—3606. 

Invest in central databases for hydrological, hydrochemical and hydrobiological data to safeguard data sets, especially long time series, and to enable model development and upscaling to the regional and continental levels. 

Subsurface simulations have actually begun to impact geochemistry, geophysics, and environmental chemistry by encouraging holistic and mechanistically detailed investigations where multiple processes and multiple reactive components are considered and monitored in complex subsurface materials. Over the next 10 years, these simulations are expected to be indispensable to the advancement of subsurface science through the systematic upscaling of processes at fundamental scales to representative parameterizations useful to field-scale models. 

Roth, K., Vogel, H.J., and Kasteel, R. The scaleway a conceptual framework for upscaling soil properties in modelling of transport processes in soils at various time and space. International workshop of EurAgEng s field of interest on soil and water. 477-490. 1999. 

Upscaling via modelling and simulation based on data from an appropriately designed and operated HT testing device. 

An extensive examination of the fracture network and mechanical data has been undertaken to determine models of the fracture characteristics of the three formations, the uncertainties in the parameterisation of the models, and the sensitivity of the upscaled flow properties to the underlying parameter variations. The methodology used to calculate effective hydraulic conductivity values and their sensitivity to the small-scale model is described in Blum el al. (2003). The study undertaken to obtain the effective hydraulic conductivity under different stress conditions and presented in Blum et al. (2003) revealed that the important parameters in modelling HM processes in the fractured rock mass are the fracture density, the mechanical (M) properties and the M property variations through the rock mass. 

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