Rate values, graphical differentiation

For differential methods, moderate mathematical efforts are required. However, the accuracy in calculations of the rate is low since numerical or graphical differentiation is used to determine the rate values from concentration vs time diagrams (Figure 10.10). 

Figure 10.10. Illustration of graphical differentiation used to determine the rate values...

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... 

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. 

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. 

In graphical form, the place where the rate of change is zero is the point where the slope of the curve is zero (Figure 4.5.1). From this we can see that an optimum point may be found graphically where the slope is zero, mathematically where the rate of change is zero (using differential calculus), or numerically, where the optimum is estimated from known data values. 

Since the replots of slopes or intercepts versus / are nonlinear, it is not possible to determine directly the values of kinetic constants from the data in Fig. 2 instead, it is necessary to apply a differential method to rate equations, in order to obtain a graphical solution (Cleland, 1967, 1979). By using the differential method, we are raising the horizontal axis in Fig. 2 ensuring that curves become hyperbola that start at the origin ... 

Differential Method In order to use the differential method of data analysis, it is necessary to differentiate the reactant concentration versus space-time data obtained in a plug-flow PBR. There are three methods of differentiation that are commonly used (i) graphical equal-area differentiation, (ii) numerical differentiation or finite difference formulas, and (iii) polynomial fit to the data followed by analytical differentiation. The aim of differentiation is to obtain point values of the reaction rate ( Ra)p at each reactant concentration Q4 or conversion xa or space time (.W/Fao), as required. All three differentiation methods can introduce some error to the evaluation of -Ra)p- Information on and illustration of the various differentiation techniques are available in the literature [23, 26]. 

After that, the user has to decide how the results should be visualized. It is possible to print the answer in the form of individual values of the desired function, an array, etc. However, the most visual output form is the graphical one. The plotting of the results is provided by the command odeplot from the graphical library plots. Figures 3.11 and 3.12 show a solution of the differential equation set, which describes the kinetics of the first-order reversible reaction B with arbitrary rate constant values. 

The rate expression is complex, so the mass- and energy-balance differential equations (6.78) and (6.80) must be solved numerically using a step-by-step procedure from given initial values corresponding to the known conditions at the inlet of each catalyst bed. A graphical integration procedure is illustrated in the upper section of Fig. 6.9. 

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