Graphical differentiation and

Three methods are commonly used to estimate this quantity (1) slopes from a plot of n versus f, (2) equal-area graphic differentiation, or (3) Taylor series expansion. For details on these, see a mathematics handbook. The derivatives as found by equal-area graphic differentiation and other pertinent data are shown in the following table ... 

Graphical Differential and Integral Logarithmic Extrapolation Graphical Differential and Integral... 

The experimental points scatter uniformly on both sides of the line. Accordingly, it can be concluded that the tested rate equation should not be rejected. The slope, k, is 0.02 min. This is only a rough estimate of the rate constant because numerical and graphical differentiations are very inaccurate procedures. The slope was also calculated by the least squares technique minimizing the sum of squares... 

If AH is plotted against m at constant 2, a graphical differentiation by the chord-area method will yield L ii as a function of composition. Alternatively, the data could be fitted to a polynomial and the derivative of that pol5momial then could be computed. Differentiation of Equation (18.12) with respect to 2 at constant m yields... 

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. 

Numerous procedures have been developed for graphical differentiation. A particularly convenient one (9), which we call the chord-area method, is illustrated using the same data (from Table A.2) to which we previously apphed numerical differentiation. It is clear from Figure A.2 that if we choose a sufficiently small temperature interval, then the slope at the center of that interval will be given approximately by A /Af. In this example, with an interval of 5°C, the approximation is good. Then we proceed to tabulate values of A °/At from 0°C, as illustrated in Table A.6 for the first few data. Note that values of are placed between the values of to which they refer, and the temperature intervals (5°C) are indicated between their extremities. Similarly, as L%°jis an average value (for example, —0.000484) within a particular region (such as 0°C to 5°C), values in the fifth column also are placed between the initial and the final temperatures to which they refer. 

The principal use of Eq. (173) is in conjunction with a similar heat dispersion equation. Unfortunately, a system of coupled nonlinear partial differential equations then has to be solved, which is very difficult even with the aid of computers. In the oxidation of sulfur dioxide. Hall and Smith (HI) found relatively good agreement between theory and experiment near the center of the reactor. Their calculations were based on the heat-dispersion equation, and they did not take detailed mass dispersion into account. Baron (B2) later solved the mass and heat dispersion equations simultaneously by a novel graphical method, and found better agreement between his calculations and the data of Hall and Smith. 

The student of thermodynamics must learn to cope with the functional, differential, and derivative relationships in (1.2)—(1.7) from a variety of formulaic, graphical, and experimental aspects. Let us briefly discuss each in turn. 

Indeed, the values of the derivatives dx/dt and dy/dt at t = 6 can readily be determined (for example, by graphical differentiation) from the experimental data. Then, the system (2.46) transforms into an ordinary system of two algebraic equations with two unknown quantities k[A2 and k[B2 ... 

Graphical Gradients and Intercepts Differentiation in Thermodynamics Equation of State for an Ideal Gas... 

The number of different functions is equal to the number of steps in the sequence. To determine the mechanism we thus have to find a (finite) number of functions which by linear superposition gives the reciprocal velocity. For a preliminary orientation it is often a good practice to find dt/dx by numerical (graphical) differentiation for different values of x in different experiments. This method has been used extensively by Bodenstein and his school. 

This indicates that the required D (c is determined by graphical differentiation of a plot for (1/2) [pa(c °) + ( >< 1°°)] °° vs-ci°°- It is n°t a serious disadvantage of this method that both Da and Dd must be determined experimentally, since, in most work, the measurement of an absorption curve is followed by the determination of the corresponding desorption. 

Using Equations 7 through 12 rates of imide formation were calculated according to Equation 6 and plotted vs. time in comparison with experimental rates obtained by graphical differentiation of curves obtained from experimental data. The result is shown in Figure 9. 

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... 

The order in which PEH, AP, and MPC formed was determined from the time dependence of product accumulation rate rations at t 0. The variation of these rations with time was evaluated by graphic differentiation [17]. 

Having in mind the possibility of inaccuracy inherent to data smoothing and graphical differentiation, it is still quite interesting to see the characteristic distributions of (Aiob bl r-i and p. Here the drase phase height Lf is approximately 35 cm. The term (A obnb)0r=i remains approximately constant in the dense phase, and drops rapidly to zero above Lf. Mass transfer seems somewhat better in the transition zone and in the jetting zone than... 

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