Numerical Problem-Solving Overview

You are given the mass in kilograms and asked to find the volume in cubic centimeters. Density is the conversion factor between mass and volume. 

Build the solution map starting with kg and ending with cm. Use the density (inverted) to convert from g to cm.  

Follow the solution map to solve the problem. Round the answer to three significant figures to reflect the three significant figures in the given quantities. 

Check your answer. Are the units correct Does the answer make physical sense  

The units of the answer are those of volume, so they are correct. The magnitude seems reasonable because the density is somewhat less than 1 g/cm therefore the volume of 60.0 kg should be somewhat more than 60.0 X 10 cm.  

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In spite of the broad use of optimization, from fundamental sciences to practical engineering design, there is no single numerical method that can guarantee a solution. There are a variety of methods available, each one capable of solving a class of problems, and the pursuit of new methods is still an active area of research. The interested reader is referred to textbooks, such as Refs. [6,7], to get a comprehensive view of the methods available and learn the theory behind the different techniques. A brief overview of some of these methods, in the context of parameter estimation in chemical kinetics, is presented in Ref. [8]. 

Figure 25.4 gives an overview of the various levels of mesoscopic/macroscopic models that are typically employed in the literature. The modeling and simulation of any physical problem involves several steps. The first step is to make a set of approximations to constmct a mathematical representation of the physical problem that can be later solved using the computational resources available. The second step is to discretize the set of equations, as the problem typically cannot be solved in continuous space and time. The third step is to use the appropriate set of numerical methods (robust, accurate, etc.) to solve the discretized equations and simulate the physical problem. The fourth step is to visualize/analyze the results to... 

As a consequence of the 3d and time-dependent nature of buoyant flows and of the current status of development of the standard turbulence models, there is currently no way to achieve reliable results by standard models on the temperature fields in the reactor sump. The only way which is nowadays often considered to give a better solution for 3d time-dependent flows is to apply Large Eddy Simulation methods (LES). Indeed, there exist already several applications of LES to reactor typical flows for an overview see Grotzbach Worner (1999), These show the tremendous potential of the LES method and that its possibilities are going far beyond those of standard Reynolds averaged turbulence models. The main problems which need to be solved for LES methods in this context are e.g. the development of more universal subgrid scale models, of boundary conditions for buoyant flows, and of numerical methods in commercial codes that fulfill LES requirements. 

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