Mathematical model, time-dependent

The complexity of the mathematics for time-dependent assessment and component models mean this type of modelling can only realistically be achieved using computer software, and its usefulness is limited to very specific scenarios where accepted rate-based probabilities cannot be applied or approximated. 

The software implemented is capable of learning, hence it will improve its performance with time. A user interface adapted to the process engineer responsible for the production allows to improve and extend the rule system of the heuristic component. The extent of the basic mathematical model clearly depends on the knowledge available about the process under consideration. It should be extended by relationships between the different quantities, e.g., the yield expressions within the balances based on stoichiometric or thermodynamic considerations whenever more detailed knowledge becomes available. Thus, it is strongly suggested to reexamine hybrid models from time to time with the aim of improving the classical mechanistic part of the model. 

The idea of using mathematical modeling for describing materials behavior under loads is well known. Some physical phenomena, which can be observed in materials during testing, have time dependent quantitative characteristics. It gives a possibility to consider them as time series and use well developed models for their analysis [1, 2]. Usually applied... 

Thus, the time-dependent BO model describes the adiabatic limit of QCMD. If QCMD is a valid approximation of full QD for sufficiently small e, the BO model has to be the adiabatic limit of QD itself. Exactly this question has been addressed in different mathematical approaches, [8], [13], and [18]. We will follow Hagedorn [13] whose results are based on the product state assumption Eq. (2) for the initial state with a special choice concerning the dependence of 4> on e ... 

For practical reasons, the blast furnace hearth is divided into two principal zones the bottom and the sidewalls. Each of these zones exhibits unique problems and wear mechanisms. The largest refractory mass is contained within the hearth bottom. The outside diameters of these bottoms can exceed 16 or 17 m and their depth is dependent on whether underhearth cooling is utilized. When cooling is not employed, this refractory depth usually is determined by mathematical models these predict a stabilization isotherm location which defines the limit of dissolution of the carbon by iron. Often, this depth exceeds 3 m of carbon. However, because the stabilization isotherm location is also a function of furnace diameter, often times thermal equiHbrium caimot be achieved without some form of underhearth cooling. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this sec tion. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. 

The effect of the disturbance on the controlled variable These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exac t, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the abihty to predict the appropriate control ac tion). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standara prac tice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. 

A numerical value is obtainable by integrating the trend curve for the flow received from the Flow Recorder (FR), from the start of the reaction to a time selected. Doing this from zero to each of 20 equally spaced times gives the conversion of the solid soda. Correlating the rates with the calculated X s, a mathematical model for the dependence of rate on X can be developed. 

The reader is encouraged to use a two-phase, one spatial dimension, and time-dependent mathematical model to study this phenomenon. The UCKRON test problem can be used for general introduction before the particular model for the system of interest is investigated. The success of the simulation will depend strongly on the quality of physical parameters and estimated transfer coefficients for the system. 

From these results, we have constructed a mathematical model (24) for the time-dependent emission which Incorporates three reactive Intermediates ... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

Any numerical experiment is not a one-time calculation by standard formulas. First and foremost, it is the computation of a number of possibilities for various mathematical models. For instance, it is required to find the optimal conditions for a chemical process, that is, the conditions under which the reaction is completed most rapidly. A solution of this problem depends on a number of parameters (for instance, temperature, pressure, composition of the reacting mixture, etc.). In order to find the optimal workable conditions, it is necessary to carry out computations for different values of those parameters, thereby exhausting all possibilities. Of course, some situations exist in which an algorithm is to be used only several times or even once. 

The nature of the mathematical model that describes a physical system may dictate a range of acceptable values for the unknown parameters. Furthermore, repeated computations of the response variables for various values of the parameters and subsequent plotting of the results provides valuable experience to the analyst about the behavior of the model and its dependency on the parameters. As a result of this exercise, we often come up with fairly good initial guesses for the parameters. The only disadvantage of this approach is that it could be time consuming. This counterbalanced by the fact that one learns a lot about the structure and behavior of the model at hand. 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

When chemicals are released in the environment, their hazard potential to human or ecological receptors depends upon the extent of contact between the receptors and the chemical. This exposure level is not only influenced by where, when and how much of the chemical is released, but also on its movement and changes in air, water, soil or biota relative to the locations of the receptors. Risk is defined as the probability of some adverse consequence in the health context, or as the probability times the extent of the consequence in the technology context. In this paper we shall examine and discuss how mathematical models are used to generate estimates of risk when more than one of the environmental media must be considered in tracing pathways connecting sources with receptors. The principal objective here is to place in perspective the... 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

However, if you extend this notion to an extreme and make 100,000 production runs of one unit each (actually one unit every 315 seconds), the decision obviously is impractical, since the cost of producing 100,000 units, one unit at a time, will be exorbitant. It therefore appears that the desired operating procedure lies somewhere in between the two extremes. To arrive at some quantitative answer to this problem, first define the three operating variables that appear to be important number of units of each run (D), the number of runs per year (n), and the total number of units produced per year (Q). Then you must obtain details about the costs of operations. In so doing, a cost (objective) function and a mathematical model will be developed, as discussed later on. After obtaining a cost model, any constraints on the variables are identified, which allows selection of independent and dependent variables. 

Quantum wave mechanics gave chemistry a new "understanding," but it was an understanding absolutely dependent on purely chemical facts already known. What enabled the theoretician to get the right answer the first time, in a set of calculations, was the experimental facts of chemistry, which, Coulson wrote, "imply certain properties of the solution of the wave equation, so that chemistry could be said to be solving the mathematicians problems and not the other way around."36 So complex are the possible interactions among valence electrons that one must either use an exact mathematical model of a... 

The modeling discussed here depends on being able to describe the entire concentration-time curve. This can only be done using a Level A IVIVC (i.e., a point-to-point relationship between in vitro release and in vivo release/absorption). In fact, the U.S. Food and Drug Administration (FDA) defines a Level A IVIVC as a predictive mathematical model for the relationship between the entire in vitro dissolution-release time course and the entire in vivo response time course. 

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