Physical modeling

Physical modeling can be classified into the following three stages in order of 

A variety of examples of the second and third categories are given in this book, and the key features of physical modeling are examined. 

Of lesser importance were listed dimensionless ambient velocity, vorticity, and atmospheric lapse rates. These dimensional considerations lead to several possible strategies for simulating mass fires. They are the following. 

Standard scaling Keep some groups constant. Usually Froude number and mass burning rate are chosen which leads to distortions in radiation, convection, turbulence and fuel bed geometry. Basically one is just looking at a buoyant plume. 

Pressure adjustment Varying pressure might preserve scaling of all but radiation parameters. Unfortunately to scale a 1 mile fire to 50 feet would requires use of 1000 atm pressure. 

Pressure and body force adjustment Varying pressure and G (centrifuge) simultaneously allows consideration of all core variables except blackbody radiation. 

Adjustment of composition and temperature of ambient atmosphere Varying pressure and temperature could permit scaling of all core variables except absorptive radiation. 

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It is very important, from one hand, to accept a hypothesis about the material fracture properties before physical model building because general view of TF is going to change depending on mechanical model (brittle, elasto-plastic, visco-elasto-plastic, ete.) of the material. From the other hand, it is necessary to keep in mind that the material response to loads or actions is different depending on the accepted mechanical model because rheological properties of the material determine type of response in time. The most remarkable difference can be observed between brittle materials and materials with explicit plastic properties. 

Elaboration of a Physical Model by Eddy Current and Application in the Characterization of Long Defects in Hollow Cylindrical Products. 

We propose a physical model, being based on the variation of the magnetic flow, allowing to determine defects characteristics. Some is the material and some is the geometry of the product, the presence of a defect allocates the flow in the straight section of the solenoid. The theoretical model proposed allows to find defects characteristics from the analysis of the impedance variation. 

In a regression approach to material characterization, a statistical model which describes the relation between measurements and the material property is formulated and unknown model parameters are estimated from experimental data. This approach is attractive because it does not require a detailed physical model, and because it automatically extracts and optimally combines important features. Moreover, it can exploit the large amounts of data available. 

Experimental investigations in attestation of the procedures of testing are performed on real test objects or on physical model that is close to the real test object. 

The physical model is thus that of a liquid him, condensed in the inverse cube potential held, whose thickness increases to inhnity as P approaches... 

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

Despite the many advantages of these physical models, they have several essential deficiencies. As the size of the structure increases, the assembly of a model becomes progressively more unmanageable and more complex. Furthermore, d.ata such as atom distances and atom angles can be determined only with difficulty, or not at all. 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

Further work is needed to build a physical model that allows prediction of the concentration effect from the primary properties of the slurry or from a limited amount of slurry testing. 

Fig. 8. The photodiode detector (a) band model where the photon generates electron—hole pairs that are separated by the built-in potential setting up a photocurrent (b) physical model for a planar diode. The passivation is typically Si02 for Si diodes, an In oxide for InSb diodes, and CdTe for HgCdTe...

Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be... 

To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reac tor is operating isothermaUy and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A B takes place. The inlet concentration of A, which we shall call Cj, varies with time. A dynamic mass balance for the concentration of A (c ) can be written as follows ... 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use or transfer functions and block diagrams. A transfer func tion can be obtained by starting with a physical model as... 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

Serth, R.W, B. Srikanth, and S.J. Maronga, Gross Error Detection and Stage Efficiency Estimation in a Separation Process, AlChE Journal, 39(10), 1993, 1726-1731. (Physical model development, parameter estimation)... 

The second classification is the physical model. Examples are the rigorous modiiles found in chemical-process simulators. In sequential modular simulators, distillation and kinetic reactors are two important examples. Compared to relational models, physical models purport to represent the ac tual material, energy, equilibrium, and rate processes present in the unit. They rarely, however, include any equipment constraints as part of the model. Despite their complexity, adjustable parameters oearing some relation to theoiy (e.g., tray efficiency) are required such that the output is properly related to the input and specifications. These modds provide more accurate predictions of output based on input and specifications. However, the interactions between the model parameters and database parameters compromise the relationships between input and output. The nonlinearities of equipment performance are not included and, consequently, significant extrapolations result in large errors. Despite their greater complexity, they should be considered to be approximate as well. 

Intended Use The intended use of the model sets the sophistication required. Relational models are adequate for control within narrow bands of setpoints. Physical models are reqiiired for fault detection and design. Even when relational models are used, they are frequently developed bv repeated simulations using physical models. Further, artificial neural-network models used in analysis of plant performance including gross error detection are in their infancy. Readers are referred to the work of Himmelblau for these developments. [For example, see Terry and Himmelblau (1993) cited in the reference list.] Process simulators are in wide use and readily available to engineers. Consequently, the emphasis of this section is to develop a pre-liminaiy physical model representing the unit. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

The models must be considered to be approximations. Therefore, the goals of robustness and uniqueness are rarely met. The nonlinear nature of the physical model, the interaction between the database and the parameters, the approximation of the unit fundamentals, the equipment boundaries, and the measurement uncertainties all contribute to the limitations in either of these models. 

Many thermodynamic quantities can be calculated from the set of normal mode frequencies. In calculating these quantities, one must always be aware that the harmonic approximation may not provide an adequate physical model of a biological molecule under physiological conditions. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

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