Mathematical modeling solution method specification

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

In this chapter we have provided an overview of mathematical modeling from inception of design through specification of solution method, production of solution, and analysis of results. Additionally, we have provided a framework for including computers, particularly current and emerging application software, as vital agents in the modeling process. 

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. 

The techniques developed earlier for solving column sections are now extended to include total columns. A mathematical model is set up which is used to derive graphical methods of solution on a Y-X diagram. Solution methods are described for different sets of specifications. 

Once the mathematical model of gas emission in coal particle is set up, how to choose the specific conditions is vital for the solution. The conditions of coal samples with different particle size are varied greatly, even the physical parameters of the same coal sample under different pressures change a lot, and bringing all conditions into the model is almost impossible, so an effective method is needed to simplify the solution (Zhou Lin 1999). Actually the gas flow in coal particle has much in common, and the dimensionless method can combine several meaningful physical parameters to reflect the common (Zhang et al. 2001, Qin et al. 2009, Qin et al. 1998), which can not... 

Velocity-Dependent Friction Research and development work on the numerical specification of contact and friction conditions may include mathematical formulation and implementation of friction models as well as adaptation of the numerical solution methods (Heisel et al. 2009 Neugebauer et al. 2011). Standard implementations may be illustrated using Coulomb s Law and the Friction Factor Law. These two basic models were modified using a stick-slip model. Using these models enables consideration of the relative sliding velocity between the tool and the workpiece. 

The discussions in the following sections will focus on the thermal-hydraulic properties and characteristics of flows in parallel channels and NCLs. The hterature on general aspects of the analytical, experimental, mathematical modeling, numerical solution methods, and computational aspects of these flows will be briefly reviewed. These aspects when associated with specific Gen IV systems will also be discussed. 

According to Hinunelblau and Bischoff [1] Process analysis is the application of scientific methods to the recognition and definition of problems and the development of procedures for their solution. In more detail, this means (1) mathematical specification of (he problem for the given physical solution, (2) detailed analysis to obtain mathematical models, and (3) synthesis and presentation of results to ensure full comprehension. ... 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

The main specificity of the lEF method is that, instead of starting from the boundary conditions as in the DPCM, it defines the Laplace and Poisson equations describing the specific system under scrutiny, here including also anisotropic dielectrics, ionic solutions, liquids with a flat surface boundary, quadrupolar liquids, and it introduces the relevant specifications by proper mathematical operators. The fundamental result is that the lEF formalism manages to treat structurally different systems within a common integral equation-like approach. In other words, the same considerations exploited in the isotropic DPCM model leading to the definition of a surface cheurge density a(s) which completely describes the solvent reaction response, are still valid here, also for the above mentioned extensions to non-isotropic systems. 

---