Development of mathematical model to describe

Experimental observations show that bituminous coals satisfy at least one important macroscopic characteristic of a cross-linked network they swell in numerous solvents without being dissolved by them even at high temperatures Cl, 6 9), unless thermal degradation or reaction occurs In the development of mathematical models to describe network behavior, crosslinks are assumed to be points (usually carbon atoms) or short bfudgeA (usually of molecular weight much smaller than Mc) whence three or more chains are initiated. 

The development of mathematical models to describe the thermochemical process occurring in a fluidized bed involves setting up the material and energy balance equations. The total process is represented in terms of a set of independent equations which are solved simultaneously to obtain such quantities as combustion efficiency, sulfur retention, oxygen utilization, oxygen and sulfur dioxide concentration profiles in the bed, etc. 

The rate of reaction is generally a function of temperature and composition, and the development of mathematical models to describe the form of the reaction rate is a central problem of applied chemical kinetics. Once the reaction rate is known. 

In efficient or critical adhesive-bonding applications, it is necessary to demonstrate that a bonded Joint will behave as required in service. Bond performance (the characteristics under the envisaged stress and environmental conditions) is assessed by a combination of analysis and testing. A iarge amount of effort has been devoted to the development of mathematical models to describe the performance of bonded joints and provide quantitative values for strength, stiffness, etc. It must be said that, despite this, it is still very difficult to produce realistic predictions of joint characteristics for all but the simplest designs and materials. It continues to be a topic of investigation. 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

Gao et al. [71, 72] developed a mathematical model to describe the effect of formulation composition on the drug release rate for hydroxypropyl methylcellulose-based tablets. An effective drug diffusion coefficient T>, was found to control the rate of release as derived from a steady-state approximation of Fick s law in one dimension ... 

Harrison and Hills [64] developed a mathematical model to describe flavour release from aqueous solutions containing aroma-binding macromolecules. Seuvre et al. [65] reported that the retention of flavouring substances (benzaldehyde, isoamyl acetate. 

Using the Equations 11.25 and 11.27 as a basis Epstein [1994] has developed a mathematical model to describe the initial rate of chemical reaction fouling. Combining Equations 11.25 and 11.27 gives... 

Wingard and Philbps developed a mathematical model to describe the effect of temperature on extraction rate using a percolation extractor as follows ... 

The aim of the present smdy is to develop a mathematical model to describe the evaporation and drying of PVP- and mannitol-water droplets including the particle formation. The results of the numerical simulations are compared with... 

Risk specialists have developed. sophisticated mathematical models to describe the methodological and scientific uncertainty in regulatory risk assessments. In this chapter, I focus on those aspects of uncertainty that are most affected by documentation practices in hazardous environments. To set the stage for this discussion, I describe the problem of uncertainty at the highest level of exigence imminent danger. 

In order to investigate the mechanisms of thermal regulation of the PCM on the heat and moisture transfer in textiles, Li and Zhu developed a mathematical model to describe the energy loss rate from the microspheres which is considered to be a sphere consisting of solid and liquid phases [18], as shown in the following equations ... 

In this work the development of mathematical model is done assuming simplifications of physico-chemical model of peroxide oxidation of the model system with the chemiluminesce intensity as the analytical signal. The mathematical model allows to describe basic stages of chemiluminescence process in vitro, namely spontaneous luminescence, slow and fast flashes due to initiating by chemical substances e.g. Fe +ions, chemiluminescent reaction at different stages of chain reactions evolution. 

It is evident from the foregoing description and diagrams shown in Fig. 1,7a, b that multipurpose batch chemical plants are more complex than multiproduct batch plants. This complexity is not only confined to operation of the plant, but also extends to mathematical formulations that describe multipurpose batch plants. Invariably, a mathematical formulation that describes multipurpose batch plants is also applicable to multiproduct batch plants. However, the opposite is not true. It is solely for this reason that most of the effort in the development of mathematical models for batch chemical plants should be aimed at multipurpose rather than multiproduct batch plants. 

Due to the complexity of insect cell/baculovirus interactions and the possibility of using low MOTs, which, in turn, increase the complexity of the process since several different population types coexist simultaneously in the bioreactor vessel, the development of mathematical models as tools for describing the system dynamics is extremely useful. 

To characterize 5-HTlA-agonist-induced hypothermia, Zuideveld et al. [554] developed a mathematical model that describes the hypothermic effect on the basis of the concept of a set point and a general physiological response model [431,555]. The model was applied to characterize hypothermic response vs. time profiles after administration of different doses of the reference 5-HT1A receptor agonists R- and S-8-OH-DPAT. 

Vogler 31) developed a mathematical model to derive semiquantitative kinetic parameters interpreted in terms of transport and adsorption of surfactants at the interface. The model was fitted to experimental time-dependent interfacial tension, and empirical models of concentration-de-pendent interfacial tension were compared to theoretical expressions for time-dependent surfactant concentration. Adamczyk (32) theoretically related the mechanical properties of the interface to the adsorption kinetics of surfactants by introducing the compositional surface elasticity, which was defined as the proportionality coefficient between arbitrary surface deformations and the resulting surface concentrations. Although the expressions to describe the adsorption process differed from one another, it was demonstrated that the time-dependent interfacial tensions mirrored the change of surface-active substances at the interface. 

The chapters presented in Part El describe development of mathematical models that can be used for interpretation of impedance measurements. These models may be regressed to data using the approaches presented in Chapter 19. A systematic approach is presented in this chapter to determine whether the model provides a statistically adequate description of the data. 

In Section 1, a sufficiently detailed review of the copolymerization theory is presented. The fundamental assumptions made in development of mathematical models capable of describing copolymerization processes are presented and discussed in view of recent studies. In Section 2, the traditional polymer characterization techniques as applied to copolymers are reviewed. Copolymer... 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

It is evident that evaluation of the effectiveness of an organic acid for a specific application will require a much better understanding of general as well as specific stress response potentials of foodborne pathogens (Ricke, 2003). Predictive microbiology may be a handy tool in achieving this via mathematical models to describe the behavior of foodborne microorganisms. The concept has developed very rapidly over the past two decades... 

---