Model, mathematical validity

The models you use to portray failures that lead to accidents, and the models you use to propagate their effects, are attempts to approximate reality. Models of accident sequences (although mathematically rigorous) cannot be demonstrated to be exact because you can never precisely identify all of the factors that contribute to an accident of interest. Likewise, most consequence models are at best correlations derived from limited experimental evidence. Even if the models are validated through field experiments for some specific situations, you can never validate them for all possibilities, and the question of model appropriateness will always exist. 

Madnia, C.K., S.H. Frankel, and P. Givi. 1992. Reactant conversion in homogeneous turbulence Mathematical modeling, computational validations and practical applications. Theoretical Computational Fluid Dynamics 4 79-93. 

Tronconi et al. [46] developed a fully transient two-phase 1D + 1D mathematical model of an SCR honeycomb monolith reactor, where the intrinsic kinetics determined over the powdered SCR catalyst were incorporated, and which also accounts for intra-porous diffusion within the catalyst substrate. Accordingly, the model is able to simulate both coated and bulk extruded catalysts. The model was validated successfully against laboratory data obtained over SCR monolith catalyst samples during transients associated with start-up (ammonia injection), shut-down (ammonia... 

Barolo et al. (1998) developed a mathematical model of a pilot-plant MVC column. The model was validated using experimental data on a highly non-ideal mixture (ethanol-water). The pilot plant and some of the operating constraints are described in Table 4.13. The column is equipped with a steam-heated thermosiphon reboiler, and a water-cooled total condenser (with subcooling of the condensate). Electropneumatic valves are installed in the process and steam lines. All flows are measured on a volumetric basis the steam flow measurement is pressure- and temperature-compensated, so that a mass flow measurement is available indirectly. Temperature measurements from several trays along the column are also available. The plant is interfaced to a personal computer, which performs data acquisition and logging, control routine calculation, and direct valve manipulation. 

The use of modelling is based on a number of assumptions. The most important relate to the similarity of the modelled set of phenomena or events or cases (here chemical compounds). Any mathematical/statistical, chemical/biological modelling approach must either be based on fundamental ab initio theory without many adjustable parameters, or be derived as a Taylor or other serial expansions of such fundamental models. In the latter case the model is valid only locally , i.e. for moderate changes in its variables. In the present context, this corresponds to having only a moderate variation in chemical structure between the compounds included in one model, as well as only a moderate variation of the biological mechanism of action of the chemical compounds. If there is a wide variation, this necessitates the use of several models, one for each structural and biological-mechanistic type of compounds. 

A mathematical programming framework for optimal model selection/validation of process data... 

A Mathematical Programming Framework for Optimal Model Selection/Validation... 

An additional experiment was chosen to verify the validity of the models and to check their predictive ability, with the condition ofxi = xa = X3 = X4= 0.5, i.e. T, = 57.5 °C, <, = 9 h, 7d = 162.5 °C and = 9 h. The model-predicted and the experimental values of the responses are shown in Table 5. The observed values are within the 95% prediction interval, and within the 95% confidence interval of the predicted ones. The observed values are in good agreement with the predicted ones therefore the three models are valid and have satisfactory predictive ability. It is also confirmed that RSM is an effective tool for the mathematical modeling of the mechanical properties of solid catalysts. 

Given values for some of the parameters and their associated experimental errors, what can be said concerning the mathematical validity of the proposed reaction model ... 

The computing problem is concerned with calculating the maximum number of unknown parameters of a proposed reaction system from available experimental data. This data can be any combination of values for constant parameters (rate and equilibrium constants) and variable parameters (concentration versus time data). Moreover, data for different variable parameters need not have the same time scale. When the unknown parameters are calculated, it is important that the mathematical validity of the proposed model be determined in terms of the experimental accuracy of the data. Also, if it is impossible to solve for all unknown parameters, then the model must be automatically reduced to a form that contains only solvable parameters. Thus, the input to CRAMS consists of 1) a description of a proposed reaction system model and, 2) experimental data for those parameters that were measured or previously determined. The output of CRAMS is 1) information concerning the mathematical validity of the model and 2) values for the maximum number of computable unknown parameters and, if possible, the associated reliabilities. The system checks for model validity only in those reactions with unknown rate constants. Thus a simulation-only problem does not invoke any model validation procedures. 

The SOLVER module is the communications link between the three numerical analysis service modules NONLIN, SIMULATOR, and CURVEFIT. SOLVER solves the equations that were chosen by SELECTOR by using (1) NONLIN — to initially bring the system to equilibrium, (2) SIMULATOR — to generate concentration data for certain unknown variable parameters and (3) CURVEFIT — to solve for unknown constant parameters and to test the mathematical validity of the proposed reaction model. The SOLVER module has been designed so that the three numerical analysis service modules are easily replacable as more advanced techniques are developed. The design of the SOLVER module is described in detail in Part 4. The modules NONLIN, SIMULATOR, and CURVEFIT are discussed in 4.2., 4.3., and 4.4., respectively. 

T 3. [Eliminate Extra Known Equilibrium Constants.] Let TFLUX be the transpose of FLUX. Call DMFGR [TFLUX (IKEK - - 1,1), NKEK. NRCT, IRANK], For IRANK least squares technique were used to solve for the unknown equilibirum constants, this step would not be necessary. However, mathematical validation of the model would not then be possible. 

Unknown rate and equilibrium constants in Type (4) and Type (5) equations are computed by the curve-fitting module CURVEFIT. In the current version of CRAMS there is a choice of two curve-fitting methods, DLLSQ 2 ) or CURFIT However, new modules can be easily incorporated 25a), DLLSQ uses the simple least-squares method to compute the unknown constants and it suffers from the unreliabilities and ambiguities common to all such methods CURFIT performs a complete analysis of the experimental data (which includes user-supplied maximum tolerances) and tests the mathematical validity of the proposed reaction model, then computes the unknown constants and their associated maximum errors Since the second choice is a superset of the calculations performed by DLLSQ, in the following discussion it is assumed that CURFIT is used. 

The verification of the proposed model is also a two step process involving (1) mathematical validation (the process of verifying that the equations are valid algebraic descriptions of the data that is used to test the model) and (2) physical validation (the process of verifying that the mathematically validate equations also meet the ph5reical conditions imposed by the model). It is not generally realized... 

In the CURFIT module the mathematical validity of those equations that represent those parts of the model that contain unknown constant parameters is tested in terms of the user s estimate of the reliability of his data. However, it should be noted that mathematical validity does not, by itself, establish the worth of the model and that CURFIT does not test for ph37sical validity of the model. This can, in general, only be performed by one who is an expert in the physical meaning of the model. 

Beeckman and Hegedus [50] determined the intrinsic kinetics over two commercial vanadia on titania catalysts. A mathematical model was proposed to compute NO and SO2 conversions and the model was validated by experimental values. Slab-shaped cutouts of the monolith and powdered monolith material were used in a differential reactor. The cutouts contained nine channels with a length of 15 cm and with a channel opening and wall thickness of 0.60 and 0.13 cm, respectively. The SCR reaction over a 0.8 wt% V2O5 on titania catalyst was first-order in NO and zero-order in NH3. 

Use a mathematical validation model that takes into account results from European test standards and others. 

Additional experiments are needed for there to be enough data (N > p) for a statistical analysis. We add an experiment to the design at the centre of the domain, which is the point furthest from the positions of the experiments of the factorial design. This will allow us to verify, at least partially, the mathematical model s validity. Therefore the solubility was determined in a mixed micelle containing... 

As we indicated earlier, this model is intended to be used to predict solubility within the experimental domain, that is to calculate the solubility in solvents whose compositions differ from those of the 4 experiments in the design. We need therefore to be assured of the model s validity. We will examine the statistical aspects of this validation later in the chapter but our approach here will be more intuitive. We note first of all that the mathematical model, with the estimates of the coefficients ... 

An example of the above consideration is the equivalence of the power model proposed by Colombo et al. (1994) and Clausen s model (1993) for emission controlled by internal diffusion in the source. The assumptions made in the latter - more physically based - model lead to a final description of the emission rate at the surface of the source which is equivalent to the description of the former - purely empirical - model. The equivalence of the models is valid when the parameter C of the empirical model takes the value 1. This can readily be seen if we compare the mathematical equations of the two models ... 

Science is in incessant evolution it grows with more precise theories and better instrumentation. The thermodynamic theories of polymers and polymeric systems move toward atomistic considerations for isomeric species modeled mathematically by molecular dynamics or Monte Carlo methods. At the same time good mean-field theories remain valid and useful—they must be remembered not only for the historical evolution of human knowledge, but also for the very practical reason of applicability, usefulness, and as tools for the understanding of material behavior. 

As in the study of any model, we assume for simplicity that our model is valid at any values of the independent variables of its constitutive equations, e.g. at all positive temperatures and densities. Again, such behaviour is not fulfilled in reality and in fact this limits the range of application of such a model (cf. difference between real material and its mathematical model in Sect. 2.3). 

Indeed, once the mathematical models are validated by classical statistical tools (e.gi, ANOVA, or analysis of the variance analysis of the lack of ht) (15). we can draw the response. surfaces representing t)ic evolution of the responses in the whole domain studied, when two factors are varying and the third one is fixed. From the different diagrams of isoresprmse curves, we can determine the influence of the different factors considered on the responses-... 

Pang, S. Mathematical modeling of kiln drying of softwood timber Model development, validation and practical application. Drying Technology 25(3) 421-431, 2007. 

Figure 4. Percentage distribution of the PEMFC (including DMFC) mathematical models for the 2005 to the present period according to the dimension of both model and validation data.

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