Mathematical modeling components

Lakestani, F., Validation of mathematical models of the ultrasonic inspection of steel components, PISC III report 6, IRC Inst. Adv. Mater., Petten, 1992. 

Distillation Columns. Distillation is by far the most common separation technique in the chemical process industries. Tray and packed columns are employed as strippers, absorbers, and their combinations in a wide range of diverse appHcations. Although the components to be separated and distillation equipment may be different, the mathematical model of the material and energy balances and of the vapor—Hquid equiUbria are similar and equally appHcable to all distillation operations. Computation of multicomponent systems are extremely complex. Computers, right from their eadiest avadabihties, have been used for making plate-to-plate calculations. 

The details of the mathematical model of these four components are given below. Drainage of free liquid in thin film ... 

Even when a total system analysis is unnecessai y, the methodology of mathematical modeling is useful, because by considering each component of a system as a block of a flow sheet, the interrelationships become much clearer. Additional alternatives often become apparent, as does the need for more equipment-performance data. 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

The Rome Air Development Command (RADC - Rome NY) provides the MIL HDBK 217 series of detailed electronics information. Early reports in this series provided failure rates for electronic components. The development of integrated circuits resulted in the approach of providing parameters for mathematical models of transistors and integrated circuits. RADC also publishes Nonelectronic Parts Reliability Data covering the failure rates of components ranging from batteries to valves. 

The MSF model (NUREG/CR-3837) is used principally to determine the level of dependence between safety systems introduced by maintenance, testing, and calibration activities. It is a mathematical model which modifies the independent failure probability of any single component by considering that a component with which it is redundant has already failed. This allows the conditional failure probabilities of redundant components to be calculated to determine the overall system failure probability. Documentation requirements are given in Table 4.5-6. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

We consider the following four-component models. Model N is depicted schematically in Figure 18.10 and the corresponding mathematical model is given by the following three ODEs ... 

Detailed quantitative analyses of the data allowed the production of a mathematical model, which was able to reproduce all of the characteristics seen in the experiments carried out. Comparing model profiles with the data enabled the diffusion coefficients of the various components and reaction rates to be estimated. It was concluded that oxygen inhibition and latex turbidity present real obstacles to the formation of uniformly cross-linked waterborne coatings in this type of system. This study showed that GARField profiles are sufficiently quantitative to allow comparison with simple models of physical processes. This type of comparison between model and experiment occurs frequently in the analysis of GARField data. 

The work of Crank [38] provides a review of the mathematical analysis of well defined component transport in homogeneous systems. These mathematical models and measured concentration profile data may be used to estimate diffu-sivities in homogenized samples. The use of MRI measurements in this way will generate diffusivities applicable to models of large-scale transport processes and will thereby be of value in engineering analysis of these processes and equipment. 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 53, in terms of the mass fraction x2 of fines, and two empirical parameters n, and n2 ... 

Firewater Reliability - A mathematical model of the ability of the firewater system to provide firewater upon demand as required by the design of the system without a component failure, e.g., a Mean Time Between Failure (MTBF) analysis. 

This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

One or more components of a mathematical model that aim to replicate the drug-handling characteristics of a proportion of the body. 

The various elements that must be linked as part of an a priori methodology for relating emissions to air quality are shown in Fig. 1, where the mathematical model provides a framework for integrating the following basic components. 

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