Graphical differentiation

Eqs. 1 to 3 relate the rate of production Rj of the balanced reaction component y to the molar amounts or their derivatives with respect to the time variable (reaction time or space time, see above). From the algebraic eq. 2 for the CSTR reactor the rate of production, Rj, may be calculated very simply by introducing the molar flow rates at the inlet and outlet of the reactor these quantities are easily derived from the known flow rate and the analytically determined composition of the reaction mixture. With a plug-flow or with a batch reactor we either have to limit the changes of conversion X or mole amount n7 to very low values so that the derivatives or dAy/d( //y,0) or dn7/d/ could be approximated by differences AXj/ (Q/Fj,0) or An7/A, (differential mode of operation), or to measure experimentally the dependence of Xj or nj on the space or reaction time in a broader region this dependence is then differentiated graphically or numerically. 

The conversion-time functions were differentiated graphically the values of the difference quotient, AU/At, were plotted logarithmically vs. time. The conversion, U, is directly proportional to the concentration of the polymer, and aU is proportional to the decrease of the monomer concentration, —A[M]. Reactions of zero order (with respect to monomer concentration) thus plotted produce a parallel to the time abscissa ... 

Construct a C versus a plot and differentiate (graphically, numerically, or by using a suitable curve-fitting routine) to obtain dC/da versus a data. 

The cumulative pore size distribution by volume is obtained by plotting Vj against r ,. This may be differentiated graphically to produce the relative pore size distribution by volume or the calculation may be carried out using the tabulated data. 

Although the stagewise model is not physically reaUstic for differential contactors, it is sometimes used. The number of equivalent theoretical stages N can be determined graphically usiag the stepwise constmction illustrated ia Figure 7. For the case where both the equihbrium and operating lines are linear, it can be shown that ... 

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. 

One other approach is to measure the rate as a function of a specified argument alone, and then differentiate the function with respect to the argument used. The differentiation can be done graphically, or by fitting an empirical function to the data (like a Fourier series) and differentiating this analytically. 

In the first one, the desorption rates and the corresponding desorbed amounts at a set of particular temperatures are extracted from the output data. These pairs of values are then substituted into the Arrhenius equation, and from their temperature dependence its parameters are estimated. This is the most general treatment, for which a more empirical knowledge of the time-temperature dependence is sufficient, and which in principle does not presume a constancy of the parameters in the Arrhenius equation. It requires, however, a graphical or numerical integration of experimental data and in some cases their differentiation as well, which inherently brings about some loss of information and accuracy, The reliability of the temperature estimate throughout the whole experiment with this... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

The power requirement can, therefore, be calculated as a function of d and the cost obtained. The total cost per annum is then plotted against the diameter of pipe and the optimum conditions are found either graphically or by differentiation as shown in Example 8.9. 

The integral heat of mixing is, of course, the quantity directly measured in the calorimetric method However, the heat change on diluting a solution of the polymer with an additional amount of solvent may sometimes be measured in preference to the mixing of pure polymer with solvent In either case, the desired partial molar quantity AHi must be derived by a process of differentiation, either graphical or analytical. 

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... 

The l -value is very similar to that found from graphical calculations k = 0.021 min . Differential kinetic analysis would be much more accurate if experiments were performed in a CSTR. The rates would then be measured directly with greater accuracy and no differentiation error would be made. Moreover, the concentration of the reactant and products could then be varied independently. 

This equation is depicted graphically in Fig. 3.9. Differentiation of the potential E with respect to a yields the following relationship at the point... 

In experimental kinetics studies one measures (directly, or indirectly) the concentration of one or more of the reactant and/or product species as a function of time. If these concentrations are plotted against time, smooth curves should be obtained. For constant volume systems the reaction rate may be obtained by graphical or numerical differentiation of the data. For variable volume systems, additional numerical manipulations are necessary, but the process of determining the reaction rate still involves differentiation of some form of the data. For example,... 

The term in parentheses on the left side of equation B may be determined from the data in several ways. The bromine concentration may be plotted as a function of time and the slope of the curve at various times determined graphically. Alternatively, any of several methods of numerical differentiation may be employed. The simplest of these is used in Table 3.1.1 where dC/dt is approximated by AC/At. Mean bromine concentrations corresponding to each derivative are also tabulated. 

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