Grey model

What is commonly understood by a fundamental approach is applying theoretically based mathematical models of necessary equipment items. Intrinsic (not falsified by processes other than a chemical transformation) kinetics of all processes are investigated, transport phenomena are studied, flow patterns are identified, and relevant microscopic phenomena are studied. It is intended to separately study as many intrinsic stages as possible and to combine results of these investigations into a mathematical model. Such a model contains only a limited amount of theory (grey models, gross models, or tendency models). Obviously, the extrapolation power of these models strongly depends on the content of theory. The model... 

Empirical grey models based on non-isothermal experiments and tendency modelling will be discussed in more detail below. Identification of gross kinetics from non-isothermal data started in the 1940-ties and was mainly applied to fast gas-phase catalytic reactions with large heat effects. Reactor models for such reactions are mathematically isomorphical with those for batch reactors commonly used in fine chemicals manufacture. Hopefully, this technique can be successfully applied for fine chemistry processes. Tendency modelling is a modern technique developed at the end of 1980-ties. It has been designed for processing the data from (semi)batch reactors, also those run under non-isothermal conditions. 

Gurden SP, Westerhuis JA, BijlsmaS, Smilde AK, Modelling of spectroscopic batch process data using grey models to incorporate external information, Journal of Chemometrics, 2001, 15, 101-121. 

Yu Zhifeng Xie Zhengwen. 2007. Improved grey model by exponential smoothing for settlement predication and its application. Central South Highway Engineering (T) 12Q-122. 

Fig. 13. Model of the growth of a nanotubule bonded to the catalyst surface, (a) Growth of a straight (5,5) nanotubule on a catalyst particle, with perimeter I5ak (b) growth of a straight (9,0) nanotubule on a catalyst particle whose perimeter is 18ak (k is a constant and the grey ellipsoids of (a) and (b) represent catalyst particles, the perimeters of which are equal to 5ak and 18a/t, respectively) (c) (5,5)-(9,0) knee, the two sides should grow optimally on catalyst particles having perimeters differing by ca. 20%.

FIG. 19 Scheme of a simple fluid confined by a chemically heterogeneous model pore. Fluid modecules (grey spheres) are spherically symmetric. Each substrate consists of a sequence of crystallographic planes separated by a distance 8 along the z axis. The surface planes of the two opposite substrates are separated by a distance s,. Periodic boundary conditions are imposed in the x and y directions (see text) (from Ref. 77). 

Figure 4.18 (a) STM image (39 x 23 nm) 02 molecules at Ag(l 10) at 65 K, illustrating the hot precursor mechanism at a coverage of 0.02. The inset shows an atomic resolution image of the silver surface and the 02 molecules as dark holes. Also shown (b) is a ball model with oxygen molecules (black) and surface silver atoms (white) and second layer silver atoms (grey). (Reproduced from Ref. 32). 

Figure 10.2 Adsorbed sulfur structures on Cu(lll). (a) Model of the (x/7 x x/7) R19° phase showing the Cu4S tetramers large grey circles are added coppers, smaller circles represent S. (b) Filtered 50 x 50 nm STM image of coexisting ( /l x y 7) R19° and complex structures, (c) 5 x 5nm STM image of domain boundary between the two phases. (Reproduced from Refs. 6 and 7).

Figure 10.8 p(4 x 1)S Ni(l 10) surface, (a) High-resolution STM image, (b, c) Top and perspective views, respectively, of model structure. The sulfur atoms are black and the copper atoms are grey with increasing depth indicated by darker colours. (Reproduced from Ref. 30). 

Figure 2-5. Geometries of the 02-bound state optimized using the active-site model (left) and an ONIOM model (right). Note the large differences in geometry of the two calculations, especially the hydrogen bonds donated to O2 in the ONIOM model (marked in grey) (Adapted from Hoffman et al. [25]. Reprinted with permission. Copyright 2004 Wiley Periodicals, Inc.)...

Figure 2-7. Origins of the increased O2 binding energy in IPNS when the protein is included in an ONIOM model. (A) A comparison of the optimized geometries from an active-site model (silver) and an ONIOM protein model (dark grey), show that the artificial structural relaxation of the active-site model is more pronounced for the reactant state than for the product state. (B) Contributions to O2 binding from the surrounding protein, evaluated only at the MM level (Adapted from Lundberg and Morokuma [26], Reprinted with permission. Copyright 2007 American Chemical Society.)...

Figure 2-9. Reaction scheme for the complete catalytic cycle in glutathione peroxidase (left). Numbers represent calculated reaction barriers using the active-site model. The detailed potential energy diagram for the first elementary reaction, (E-SeH) + H2O2 - (E-SeOH) + H2O, calculated using both the active-site (dashed line) and ONIOM model (grey line) is shown to the right (Adapted from Prabhakar et al. [28, 65], Reprinted with permission. Copyright 2005, 2006 American Chemical Society.)...

Figure 2-12. (A) Overlay of the initially optimized 4req structure (white) with the TS structure (dark grey). The displayed atoms include the QM part and fragments of the MCA substrate (treated by MM). (B) Energy profile and stationary points for homolysis of the Co—C5 bond in MCM calculated using the 4req model (Adapted from Kwiecien et al. [29]. Reprinted with permission. Copyright 2006 American Chemical Society.)...

Figure 22 Schematic representation of proposed models for the fibril formation in the cases of pH 3.3 and 7.5. (A) hCT monomers in solution (B) a homogeneous association to form the a-helical bundle (micelle) (C) a homogeneous nucleation process to form the P-sheet and heterogeneous association process (D) a heterogeneous fibrillation process to grow a large fibril, a-helix, antiparallel p-sheet, and parallel p-sheet forms are shown by a box, drawn by dark grey and grey, respectively. From Ref. 163 with permission.

Fig. 2 Mechanically oriented bilayer samples as a membrane model for ssNMR. (a) Illustration of the hydrated lipid bilayers with MAPs embedded, the glass supports, and the insulating wrapping, (b) A real sample consists of 15 stacked glass slides, (c) Schematic solid-state 19F-NMR lineshapes from an oriented CF3-labelled peptide (red), and the corresponding powder lineshape from a non-oriented sample (grey), (d) Illustration of typical orientational defects in real samples - the sources of powder contribution in the spectra...

Fig. 8.29. Europium to iron ratios plotted against metallicity [Fe/H] according to the model of supernova-induced star formation, after Tsujimoto, Shigeyama and Yoshii (1999). Grey scales represent predicted stellar surface densities in the ([Fe/H],[Eu/Fe]) plane convolved with a Gaussian with o = 0.2dex for Eu/Fe and 0.15 dex for Fe/H, and symbols show observational data from various authors. The inset shows the unconvolved predictions.

Fig. 2.2. Two generation models of coherent optical phonons, (a), (c), (e) impulsive stimulated Raman scattering (ISRS). (b), (d), (f) displacive excitation of coherent phonons (DECP). Graphs (e) and (f) display the time evolution of the driving force (grey areas) and that of the displacement (solid, curves) for ISRS and DECP, respectively...

Figure 2. Example of a critical load function for S and N defined by the CLmaxS, CLminN, CLmaxN and CLnutN. Every point of the grey-shaded area below the critical load function represents depositions of N and S, which do not lead to the exceedance of critical loads (UBA, 1996 Modelling and Mapping Manual, 2004).

Figure 11.15 Four different representations of the structure of -hexane, C6H14. (a) is a three-dimensional representation showing all the atoms (b) reduces the structure to the carbon backbone, in which the hydrogens (three at each end and two at each joint ) are assumed to be present (c) is a two-dimensional representation and (d) is a space-filling model in which carbons are grey and hydrogens are white.

Figure 9. Mass - radius relations for compact star configurations with different EoS purely hadronic star with HHJ EoS (long-dashed), stable hybrid stars for HHJ - INCQM EoS with 2SC (solid) and without 2SC phase (dash-dotted) for the Gaussian formfactor. We show the influence of a tiny variation of the coupling constant Gi by the filled grey band. The difference between the models 2SC and 2SC corresponds to a shift in the bag function (see Fig. 8) 3 MeV/fm3. For comparison, observational constraints on the compactness are given from the "small compact star RX J1856.5-3754 and from the high surface redshift object EXO 0748-676 which can both be obeyed by our hybrid star EoS.

Irigoien X (2006) Reply to Horizons Article Castles built on sand dysfunctionality in plankton models and the inadequacy of dialogue between biologists and modellers Flynn (2005). Shiny mathematical castles built on grey biological sands. J Plankton Res 28 965-967 Irigoien X, Flynn KJ, Harris RP (2005) Phytoplankton blooms a loophole in microzooplankton grazing impact J Plankton Res 27 313-321... 

Fig. 4. Complete models of human reduced haemoglobin (left) and horse oxyhaemoglobin (right). The inclinations of the haem groups in the reduced form are not known their po.sitions are indicated b - balls instead of the grey-discs used in the o.xyhaemoglobin mocUd.

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