Graphical or Numerical Methods

If analytic equations for the heat capacities of reactants and products are unavailable, we still can carry out the integration required by Equation (4.78) by graphical or numerical methods. In essence, we replace Equation (4.79) by the expression 

Find the enthalpy of formation of ethyl alcohol in the International Critical Tables and the National Bureau of Standards tables (Table 4.2). Compare the respective values. Compare each of these with the value obtained from the Thermodynamic Research Center tables (Table 4.5) when combined with a value of the enthalpy of vaporization. 

According to Schwabe and Wagner [19], the enthalpies of combustion in a constant-volume calorimeter for fumaric and maleic acids are —1337.21 kJ mol and —1360.43 kJ mol respectively, at approximately 25°C. 

Standard enthalpies of formation of some sulfur compounds [Reprinted with permission from Ref. (20). Copyright 1958 American Chemical Society.] are listed below, together with that for S g) from tables of the National Bureau of Standards  

From mass spectrometric experiments [22], it is possible to compute a value of 

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The integration indicated by Equation (11.21) then is carried out in two steps. From approximately 20 K up, graphical or numerical methods can be used (see Appendix A). However, below 20 K, few data are available. Therefore, it is customary to rely on the Debye equation in this region. 

While, in a few special cases, there are analytical solutions [8] for the simultaneous equations (72) and (79), these cire not always easy to use and recourse is generally made to graphical or numerical methods. 

The information required here is not concentration versus time, but rate of reaction versus concentration. As will be seen later, some types of chemical reactors give this information directly, but the constant-volume, batch systems discussed here do not [ What does it profit you, anyway —F. Villon], In this case it is necessary to determine rates from conversion-time data by graphical or numerical methods, as indicated for the case of initial rates in Figure 1.25. In Figure 1.27 a curve is shown representing the concentration of a reactant A as a function of time, and we identify the two points Cai and Ca2 for the concentration at times q and t2- The mean value for the rate of reaction we can approximate algebraically by... 

Model parameters for second-order systems which include time delays can be estimated using graphical or numerical methods. A method due to Smith (1972) utilizes a model of the form... 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

In Figure 3.11, we exclude Ihe use of differential methods with a BR, as described in Section 3.4.1.1.1, This is because such methods require differentiation of experimental ct(t) data, either graphically or numerically, and differentiation, as opposed to integration, of data can magnify the errors. 

The ratio of the fugacity/2 at the pressure P2 to the fugacity/i at the pressure Pj can be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as Pi approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. 

This integral may be evaluated either by splitting into partial fractions, or by graphical or numerical means. Using the method of partial fractions, we obtain after some fairly lengthy manipulation ... 

In the determination of optimum conditions, the same final results are obtained with either graphical or analytical methods. Sometimes it is impossible to set up one analytical function for differentiation, and the graphical method must be used. If the development and simplification of the total analytical function require complicated mathematics, it may be simpler to resort to the direct graphical solution however, each individual problem should be analyzed on the basis of the existing circumstances. For example, if numerous repeated trials are... 

Groundwaters are also classified according to the weight or molar ratio between cations and anions. To express the essential data on the water quality, graphic and numerical methods are employed [18-22]. 

A flexible rubber pipe is formed from an ideal rubber with tensile modulus E = 1.5 MPa. The external pipe diameter initially is 30 mm and the wall thickness is 3 mm. It is filled with a fluid pressurized to 50 kPa. Find the new pipe diameter and wall thidc-ness. Assume the axial component of internal pressure is carried 1 rigid pipe connectors the pipe itself therefore carries no axial stress. (You will find this problem has no analytical solution. Use either a graphical method or numerical method (with microcomputer or programmable calculator) to obtain the solution.)... 

Since the pitot tube measures velocity at one point only in the flow, several methods can be used to obtain the average velocity in the pipe. In the first method the velocity is measured at the exact center of the tube to obtain y ,. Then by using Fig. 2.10-2 the y , can be obtained. Care should be taken to have the pitot tube at least 100 diameters downstream from any pipe obstruction. In the second method, readings are taken at several known positions in the pipe cross section and then using Eq. (2.6-17), a graphical or numerical integration is performed to obtain... 

Thus the reflection coefficients of the two membranes can be determined by means of graphical or numerical derivative of the curve obtained with couples of Ap and ATI values producing the same volumetric flow in experiments with C- or C3 constants respectively for positive or negative values of the flow. The values obtained by this method, although estimated by extrapolation to infinite volumetric flow, are rather accurate because the curve obtained by derivation is limited between 0 and 1 and thus becomes sufficiently flat for low, experimentally accessible, flows. 

The problems of this collection require numerical or graphical or sometimes analytical methods of solution. There is a large number of books and software on these topics. An outline with examples of these methods is the aim of this chapter. It is expected that the student will have access to some equivalents of the software used here for the solved problems and listed subsequently. Some of the work can be done with a programmable calculator, but not as easily as on a PC. 

Experimental data of thermod5mamic importance may be represented numerically, graphically, or in terms of an analytical equation. Often these data do not fit into a simple pattern that can be transcribed into a convenient equation. Consequently, numerical and graphical techniques, particularly for differentiation and integration, are important methods of treating thermodynamic data. 

From the design equation for a first-order reaction, eqn. (11), it follows that the reaction time is equal to the area under the curve of l/fe(l — Xa) plotted against Xa- This integral may be obtained graphically by counting squares or by a numerical method. 

Numerical analysis is important in digital-computer work from another viewpoint. Sometimes it is necessary to express complex functional relationships in a simpler form. Occasionally relationships may be given in a graphical or tabular form not directly suitable for processing on digital equipment. In these situations numerical methods for curve fitting and interpolation are techniques which will necessarily be employed. 

Equation 4.51 is an integral equation that can be used to determine D(c ) by a graphical construction or numerical solution. The derivative required in Eq. 4.51 is provided by the measured concentration profile at time t and the integration is performed on the inverse of c x) [6]. However, this historically important method is only moderately accurate, and it would be preferable to obtain diffusion profiles for various assumed diffusivities as a function of concentration by computation. D(c) could be deduced by fitting calculated results for a parametric representation of D(c) to an experimentally determined diffusion profile. 

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