A Worked Example

Using path integral methods, Feynman [2] showed that the Feynman-Hibbs form for the harmonic oscillator free energy is 

We will finish the example with a couple of points which will prove useful later in the chapter. If we insert the harmonic oscillator potential into the temperature correction formula (11.2), we get 

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Galerkin finite element procedure - a worked example... 

The praetieal utilization of linear reetifieation is demonstrated later through a worked example. Fitting statistieal distributions to sample data using the linear reetifieation method ean be found in Ayyub and MeCuen (1997), Edwards and MeKee (1991), Kottegoda and Rosso (1997), Leiteh (1995), Lewis (1996), Metealfe (1997), Misehke (1992), Rao (1992), and Shigley and Misehke (1989). 

The varianee equation provides a valuable tool with whieh to draw sensitivity inferenees to give the eontribution of eaeh variable to the overall variability of the problem. Through its use, probabilistie methods provide a more effeetive way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer ean quiekly foeus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the varianee eontribution of eaeh of the design variables in a stress analysis problem. 

It is not proposed here to provide instruction in the techniques of standard costing but merely to illustrate the uses to which certain of the methods can be put. This can best be shown by the use of a worked example. The following is the statement for a service department supplying a factory, which uses both fuel and materials. 

The two-layer model is being progressively updated as fresh experimental results and correlations become available. The most satisfactory starting-point for anyone wishing to use the model to calculate pressure gradients for flow of solids-liquid mixtures in a pipeline is the text of SHOOK and Roc.o(52) which includes a worked example. However, there are many pitfalls to be avoided in this area, and there is no substitute for pracucal experience gained by working in the field. 

Further detailed discussion of Kern s method together with a worked example is presented in Volume 6. 

Solution The coding is left to the reader, but if you really need a worked example of the Runge-Kutta integration, check out Example 6.4. The following are detailed results for At= 1.0, which means that only one step was taken to reach the answer. 

This mass balance equation shows that material that is initially at radial position rin will move to radial position r for some downstream location, >0. A worked example of radial velocities and curved streamlines is given in Chapter 13, Example 13.10. 

The trade-offs between direct calibration and standard addition are treated in Ref 103. The same recovery as is found for the native analyte has to be obtained for the spiked analyte (see Section 3.2). The application of spiking to potentiometry is reviewed in Refs. 104 and 105. A worked example for the application of standard addition methodology to FIA/AAS is found in Ref 106. Reference 70 discusses the optimization of the standard addition method. 

A recursive modelling procedure will let the chemist use formulated products as a raw materials. In this paper, we illustrate this use of recursion, and provide a working example that highlights some of the complexities encountered when using this technique. 

D.A. Rudd, L.A. Apuvicio, J.E. Bekoske and A.A. Trevino, The Microkinetics of Heterogeneous Catalysis (1993), American Chemical Society, Washington DC]. Ideally, as many parameters as can be determined by surface science studies of adsorption and of elementary steps, as well as results from computational studies, are used as the input in a kinetic model, so that fitting of parameters, as employed in Section 7.2, can be avoided. We shall use the synthesis of ammonia as a worked example [P. Stoltze and J.K. Norskov, Phys. Rev. Lett. 55 (1985) 2502 J. Catal. 110 (1988) Ij. 

A worked example of the detrital-component correction and error propagation for the data of Figure 4 is given in Appendix 11. 

Before we leave you to start on the problems, we feel that it is vital to present a worked example. Though of course you can work as you like, we highly recommend that you try to follow a relatively standard procedure, which will allow you to put together your facts and deductions in a systematic way and thus make it easier for you to arrive at the correct solution in a minimum time. 

As a working example consider the thermoelectric system consisting of two reservoirs of energy and electrons, in which the temperatures are 7 and T2 and the electrochemical potentials are fj, 1 and fj,2-... 

We have introduced into each chapter a number of worked examples which we believe are essential to a proper understanding of the methods of treatment given in the text. It is very desirable for a student to understand a worked example before tackling fresh practical problems himself. Chemical Engineering problems require a numerical answer, and it is essential to become familiar with the different techniques so that the answer is obtained by systematic methods rather than by intuition. ... 

The Glucose Sensor a Worked Example of Voltammetric Analysis... 

Cooper and Jeffreys [3] give a worked example in which an endothermic reaction carried out with various heating rates is considered in some detail. 

A worked example of.the use of Leung s method is given in 6.5 below. 

Relief sizing for untempered hybrids is similar to that for gassy systems, for which a worked example is given in section 7.6)... 

A value of G has been calculated using the Omega method. This is shown as a worked example in A8.5 and G is calculated as 3792 kg/m2s at the relief pressure of... 

This example is also used as a worked example for level swell calculations in A3.3.6. A reactor of volume 3.6 m3 contains 2610 kg of reactants under worst case runaway conditions. The relief pressure is 5.5 bara and the maximum accumulated pressure is 7.0 bara. 

The process looks very complicated but it is very structured and mechanical and is best shown by a worked example. The example chosen to demonstrate the... 

Figure 2.8 Factor plots for a worked example of the Taguchi methodology.

There are many other options to select the dimensionality of PLS models. Unfortunately, we should not expect them to yield the same results and, therefore, some consensus should be reached. The problem can be more difficult whenever calibration sets cannot be deployed according to an experimental design (e.g. in industrial applications see [38] for a worked example). 

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