Mathematical modelling equations

Develop a method that finds the solution of the mathematical model equations. The method may be analytical or numerical. Its complexity needs to be understood if we want to monitor a system continuously. Whether a specific model can be solved analytically or numerically and how, depends to a large degree upon the complexity of the system and on whether the model is linear or nonlinear. 

A better understanding of the behavior of FCC units can be obtained through mathematical models coupled with industrial verification and cross verification of these models. The mathematical model equations need to be solved for both design and simulation purposes. Most of the models are nonlinear and therefore they require numerical techniques like the ones described in the previous chapters. 

A mathematical model equation is preferred if an equation is available that yields a quantitatively accurate description of the data within experimental error. 

The present chapter is not meant to be exhaustive. Rather, an attempt has been made to introduce the reader to the major concepts and tools used by catalytic reaction engineers. Section 2 gives a review of the most important reactor types. This is deliberately not done in a narrative way, i.e. by describing the physical appearance of chemical reactors. Emphasis is placed on the way mathematical model equations are constructed for each category of reactor. Basically, this boils down to the application of the conservation laws of mass, energy and possibly momentum. Section 7.3 presents an analysis of the effect of the finite rate at which reaction components and/or heat are supplied to or removed from the locus of reaction, i.e. the catalytic site. Finally, the material developed in Sections 7.2 and 7.3 is applied to the design of laboratory reactors and to the analysis of rate data in Section 7.4. 

Even apart from this, the mathematical model (equation 6.2) was not a full second-order one, as only 2 interactions, X X2 and X4X5, were postulated. 

Estimating of Parameters in the Mathematic Model. Equation (17) was fitted to the data obtained by sampling method 2. Using a minimization technique, a least squares fit was obtained and parameters estimated for the model. 

Developing mathematical model equations for production processes... 

Prediction Based on the estimated values of the state variables at a given time (x t-i/jt-i)> the states for the next time (Xk/k- ) are calculated using the mathematical model (Equations 8.2-8.5). This gives the first term of the right-hand side of Equation 8.1. In addition, the covariance matrix of the estimation error (Pjt/jt-i). which is needed to determine K, is calculated (for details see Jazwinski (131]) ... 

Therefore, research focused on the development of new membrane materials and up-scaling for production is still important. In that direction, and in order to understand better the membrane behaviour of PEO-PBT copolymers, the CO2 permeability (Table 12.1) were fitted to polynomial mathematical models (Equation 12.2), and plotted versus the molecular weight (M ) of the PEO block, as shown in Figure 12.2. 

A mathematical model (Equation [21.1]) was proposed by Azrague et al. (2007) for a system in which a photocatalytic reactor was combined with... 

Nonlinearity of mathematical model equations results from reactions with orders higher than tmity or particular boundary terms (e.g., adsorption isotherms). 

If the numerical model is consistent and stable, then the solution to the numerical model equations should approach the solution to the mathematical model equations as the element size approaches zero. This means that convergence is obtained in the numerical solution. Due to the complexity of the fuel cell models described above, it is very difficult to analytically prove uniqueness, crmsistency, stability, and convergence. It is also difficult to make error estimates, even if the solution to the numerical equations does cmiverge. 

The Equilibrium Theory of chromatography is a very powerful tool to study and understand the dynamics of chromatographic columns for single component, binary and multi-component systems, whose retention behavior is described by any type of isotherm. The mathematical model equations are solved using the method of characteristics, and in the case of the Langmuir isotherm one finds out... 

Mathematical model-equations along with computer techniques are valuable tools in searching for optimal experimental conditions and effective enzymatic action. The study of... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

In this section we consider the boundary value problem for model equations of a thermoelastic plate with a vertical crack (see Khludnev, 1996d). The unknown functions in the mathematical model under consideration are such quantities as the temperature 9 and the horizontal and vertical displacements W = (w, w ), w of the mid-surface points of the plate. We use the so-called coupled model of thermoelasticity, which implies in particular that we need to solve simultaneously the equations that describe heat conduction and the deformation of the plate. The presence of the crack leads to the fact that the domain of a solution has a nonsmooth boundary. As before, the main feature of the problem as a whole is the existence of a constraint in the form of an inequality imposed on the crack faces. This constraint provides a mutual nonpenetration of the crack faces ... 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

At times, it is possible to build an empirical mathematical model of a process in the form of equations involving all the key variables that enter into the optimisation problem. Such an empirical model may be made from operating plant data or from the case study results of a simulator, in which case the resultant model would be a model of a model. Practically all of the optimisation techniques described can then be appHed to this empirical model. 

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. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

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