Mathematical modelling notation

The front-end parser allows user-friendly interaction and permits the inclusion of nonlinear constraints without recompilation. It employs a set notation which allows for writing the mathematical model in compact expressions. 

Introducing now the notation x/ (i/2) = ihx — hx/2, the discrete mathematical model of the grinding mill-classifier circuit, shown schematically in Figs. 1 and 2, is formed by the following set of recurrence algebraic equations. 

The physical system and the associated notation are depicted in Fig. 9. The assumptions made to obtain the mathematical model were as follows (i) adsorbent particles and capsules are spherical in shape (ii) adsorbent particles... 

T he mathematical model for a steady state equilibrium stage separation process consists of a large set of simultaneous nonlinear equations which must be solved to determine the phase flow rates, the stage temperatures, and the phase composition. A matrix notation was previously presented (1, 2) which permits writing the equations in a concise form... 

General mathematical model. Considering that all we have is a mass transfer phenomenon, then, in such a system, the solid concentration changes in each plant device. With the considered coordinates system and after the notations given in Fig. 3.8, we can write the mathematical model of the filter unit as a particularization of the flow equations and the solid transport equation ... 

The notion of model parameters defines one or more numerical values that are contained as symbolic notations in the mathematical model of a process. These numerical values cannot be obtained without any experimental research. In reality, the most important part of experimental research is dedicated to the identification of the models parameters. Generally, all the experimental works, laboratory... 

The following notations are used in developing the mathematical model of order scheduling discussed in this chapter ... 

For a given position of the interfaces 5, the solution of the mathematical model (Eqs. 15, 16, and 17) for the velocity u(xo) at a given point Xo can be obtained by means of the boundary integral formulation. The equations are non-dimensionalized and, for simplicity, the same notations for the dimensionless velocity, pressure, space, and time variables are used as those for the corresponding dimensional variables from the previous equations ... 

As the system of equations is solved, the mathematical model is formulated. Models are expressed in terms of mathematical symbols and notation that represent objects or systems and the relationships between them. Gomputer software, such as Maplesoft, Mathematica, MATLAB, and Excel, facilitates the process. 

Mathematical models provide a convenient and compact way of expressing the behavior of a process as a function of process physical parameters and process inputs. The same mathematical model can be expressed in several ways for example, a continuous-time model based on a differential equation can be converted to a discrete-time system, or it can be transformed to a different type of independent variable altogether (e.g., Laplace transforms or z transforms). Transform models generally feature a simplified notation, which greatly facilitates analysis of comphcated systems. 

The starting point of the study is representing the problem as a mathematical model. In this section, therefore, we formulate the problem as a mixed integer linear programming model. First, notation for the formulation is presented as follows ... 

In this section, we present the mathematical model with aU notations indices, parameters, decision variables, objective functions, and constraints as follows ... 

In formulating the detailed mathematical model, we follow the notation of Papageorgiou et al. (2001) while adding a stochastic dimension to the problem in order to account for the uncertain clinical trials outcomes. The time horizon used in our model is discretised into time intervals of equal duration. Startup and shutdown periods are considered to be negligible compared to the duration of each time interval. 

To illustrate mathematical modeling for column material-balance control, let us first lose the conventional column of Figure 14.1. The feed, Wp, is split by the column into two parts top product, Wp, and bottom product, Wp. It is assumed that vent losses overhead are negligible. It is further assumed that the heat-transfer dynamics of both the condenser and the reboiler are negligible this will be true for most columns. Let us start at the column base and work up. For convenience the equations are written in Laplace transform notation. 

The mathematical model is valid in any dimension, in particular, for two and three dimensional data. For a dimension independent notation, it is convenient to use the multi-index notation. A d-tuple of non-negative integers... 

Kinetic Model Establishment. The notation of all the reactions described in Figure 3.2B is suimnarized below to facilitate the establishment of the mathematical kinetic model ... 

We have chosen to use the x,-type notation because it is consistent with the mathematical notation used in both linear models and matrix least squares [Neter, Wasserman, and Kutner (1990)]. However, both systems are in use today. For that reason, in this chapter we will also use the classical notation, and will use it interchangeably with the x,-type notation. 

It must be ensured that the conceptual model is correctly translated into the mathematical notation, and that in turn into the computer code. Efforts to check for mistakes (test runs of example data sets, inspection of intermediate... 

The chapter begins with a short introduction to the appropriate mathematical handling of multiwavelength absorption data sets. We demonstrate how matrix notation can be used very efficiently to describe the data sets acquired in such investigations. Subsequently, we discuss in detail the two core aspects of the model-based fitting of kinetic data ... 

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